High performance sealed-gap capacitive microphone

ABSTRACT

Some preferred embodiments include a microphone system for receiving sound waves, the microphone including a back plate, a radiation plate, first and second electrodes, first and second insulator layers, a power source and a microphone controller. The radiation plate is clamped to the back plate so that there is a hermetically sealed circular gap between the radiation plate and the back plate. The first electrode is fixedly attached to a side of the back plate proximate to the gap. The second electrode is fixedly attached to a side of the radiation plate. The insulator layers are attached to the back plate and/or the radiation plate, on respective gap sides thereof, so that the insulator layers are between the electrodes. The microphone controller is configured to use the power source to drive the microphone at a selected operating point comprising normalized static mechanical force, bias voltage, and relative bias voltage level. A radius and height of the gap, and a thickness of the radiation plate, are determined using the selected operating point so that a sensitivity of the microphone at the selected operating point is an optimum sensitivity for the selected operating point.

CROSS REFERENCE

The present application is a non-provisional of, and claims priority to,U.S. Provisional Pat. App. No. 62/477,510, filed on Mar. 28, 2017; andis a non-provisional of, and claims priority to, U.S. Provisional Pat.App. No. 62/614,897, filed on Jan. 8, 2018; and is a non-provisional of,and claims priority to, U.S. Provisional Pat. App. No. 62/616,424, filedon Jan. 11, 2018; all of which are incorporated herein by reference.

BACKGROUND

The present application relates to capacitive microphones with a sealedgap between the capacitor's conductive plates, and more particularly tocapacitive Micro-machined Electro-Mechanical Systems (MEMS) microphoneswith a sealed gap for receipt of air-mediated sound.

Note that the points discussed below may reflect the hindsight gainedfrom the disclosed innovative scope, and are not necessarily admitted tobe prior art.

Microphones in consumer devices generally comprise pressure compensatedMEMS microphones and pressure compensated electret microphones. Anoverview of pressure compensated microphones—that is, microphones whichdo not have a sealed gap—is provided below.

FIG. 1 schematically shows a cross-section of an example of a pressurecompensated MEMS microphone 100. As shown in FIG. 1, a pressurecompensated MEMS microphone 100 comprises an acoustic sensor 102fabricated on a semiconductor substrate 104, the acoustic sensor 102comprising a moveable, suspended membrane 106 (a vibrating plate) and afixed sensor back plate 108. The back plate 108 is a stiff structurecomprising perforations 110 that allow air to easily move through theback plate 108. Both the membrane 106 and the back plate 108 areconnected to the substrate 104. The membrane 106 is located between theback plate 108 and the substrate 104, with a cavity 112 (a “gap”)between the membrane 106 and the back plate 108. The perforations 110enable pressure compensation of the gap 112, that is, they equalize thepressure on each side of the back plate 108. The membrane 106 issuspended over a front chamber 114 formed in the substrate 104.

The vibrating plate in a microphone can be called a membrane or aradiation plate, depending on the ratio between the radius and thicknessof the membrane or radiation plate, as further described with respect toFIG. 3.

The substrate 104 is mounted on a carrier 116, which can be, forexample, a lead frame or a printed circuit board. There is also a backchamber 118, which is surrounded by the carrier 116 and an enclosure 120(e.g., a metal casing). An integrated circuit 122 for chargingelectrodes attached to the membrane 106 and the back plate 108, and forthe interpreting the signal produced by the acoustic sensor 102, iscoupled to the membrane 106 and the back plate 108 by wire bonds 124. Asoldering pad 126 coupled to the integrated circuit 122 enables externalinput to and output from (e.g., power and signal, respectively) themicrophone 100.

The membrane 106 is a thin solid structure made of a compliant (notstiff) material, such as a perforated solid material suitable formicromachining, that flexes in response to changes in air pressurecaused by sound waves passed by the perforations 110 in the back plate108. The membrane 106 does not fully seal the gap 112. Also,perforations in the membrane 106 (not shown) increase the membrane's 106responsiveness to air-mediated sound waves by reducing membrane 106stiffness (increasing flexibility), and by helping to equalize pressureon both sides of the membrane 106 (the side facing the back plate 108and the side facing the substrate 104). As described above, theperforations 110 in the back plate 108 enable pressure compensation ofthe gap 112. In pressure compensated MEMS microphones 100 (and similarlyin pressure compensated electret microphones 200, described below), theair pressure in the gap 112 is equal to the ambient static pressure,that is, the atmospheric pressure (thus the description “pressurecompensated”). A pressure compensated gap 112 enables a more flexiblemembrane 106, because a static pressure difference between thegap-facing and substrate-facing sides of the membrane 106 is reduced.This means that there is effectively no static force against themembrane 106 due to air pressure.

The “ambient” is the medium (acoustic environment) through whichacoustic waves are conducted to intersect a membrane, causing themembrane to vibrate, resulting in a signal being emitted from themicrophone. For example, in microphones included in smartphones, therelevant ambient will generally be the atmosphere (air). As used herein,an “airborne” microphone is defined as a microphone for which theprimary intended ambient is air.

FIG. 2 schematically shows a cross-section of an example of a pressurecompensated electret microphone 200. An electret is a stable dielectricmaterial with a permanently embedded stable electric dipole moment—thatis, a permanently polarized piece of dielectric material. An electretmicrophone is a type of electrostatic capacitor-based microphone whichuses an electret, and can thereby avoid using a polarizing power supply(used in a MEMS microphone 100 to apply charge to electrodes).

As shown in FIG. 2, an electret microphone 200 comprises an acousticsensor 202, which in turn comprises an electret membrane 204 (e.g., apolymer electret membrane 204). A front chamber 206 is located on afront chamber 206 side (a first side) of the electret membrane 204. Thefront chamber 206 side of the electret membrane is electroded, and isclamped to a metal washer 208 at the electret membrane's 204 rim. Theelectret membrane 204 is separated from a back plate 210 to create a gap212 on a gap 212 side (a second side) of the electret membrane 204. Aconstant gap 212 height is maintained by, for example, plastic washers214. The back plate 210 comprises perforations 216 so that the gap 212is pressure compensated. An amplifying transistor 218 is fixedly coupledto a carrier 220 (e.g., a lead frame or printed circuit board), and theamplifying transistor's 218 gate pin is coupled by a wire 222 to theback plate 210. The connection between the amplifying transistor 218 andthe back plate 210 conveys received signal from the acoustic sensor 202to the amplifying transistor 218. The amplifying transistor 218interprets the signal produced by the acoustic sensor 202. The carrier220 is coupled to the back plate 210 by a casing 224 (e.g., plasticcasing). The carrier 220 is also fixedly coupled to a housing 226 (e.g.,a metal housing), which holds the carrier 220, the casing 224, and theacoustic sensor 202. This coupling also electrically connects theelectret membrane 204 and a source lead 228 of the amplifying transistor218. A hole 230 in the housing 226, located proximate to the frontchamber 206, gives acoustic waves access to the electret membrane 204.The hole 230 and the front chamber 206 are covered by a dust cover 232,which does not seal the electret microphone 200. That is, air, as wellas humidity and other contaminants, can access the interior of theelectret microphone 200. Contamination can be mitigated, but notprevented, by the dust cover 232. The transistor 218 is located in aback chamber 234. The back chamber 234 is also proximate to the backplate 210 on a side of the back plate 210 distant from the gap 212. Tomaintain pressure compensation, the back chamber 234 is not sealed.Access to the source lead 224 and a drain lead 236 of the amplifyingtransistor 218 are provided at an outer surface of the carrier 220 (asurface distant from the back chamber 234) to enable external electricalconnections for signal acquisition.

MEMS microphones 100 and electret microphones 200 detect sound byplacing a fixed charge across the gap 112, 212, and measuring voltagevariations caused by changes in the capacitance between the membrane106, 204 and the back plate 108, 206 as a result of the membrane 106,204 flexing in response to sound waves. MEMS microphones 100 apply thefixed charge using a bias voltage, and electret microphones 200 induce afixed charge using an electret.

Typically, MEMS microphones 100 used in mobile phones are biased at 10volts to 14 volts DC, generated using voltage doubler circuits toproduce the appropriate voltage from a battery supply outputting 1.8volts to 3.6 volts.

Typical electrets used in microphones are made of dielectric materialssuch as polymers used as membrane 204 material, or silicon oxide orsilicon nitride in the back plate 210. Electrets can trap electricalcharge in their bulk material or on their surface. Circuits including anelectret are generally terminated using a terminating impedance. Whenthe surfaces of an electret layer are properly electrically terminated,the trapped charge can yield, for example, a total charge correspondingto (which can be modeled as) a bias voltage of 150 to 200 voltspolarizing the gap 212.

As discussed, pressure compensation means that the gap is open toambient air in order to equalize gap pressure with ambient atmosphericpressure. A pressure compensated gap is therefore vulnerable tocontamination by dirt, humidity or other foreign matter carried by theair that moves to and through it. Contamination of the gap cancompromise microphone performance due to clogged gap vents, back plateperforations, and/or membrane holes, which cause noise. Membrane holecontamination reduces membrane compliance, which corresponds to a lossin microphone sensitivity. Also, material buildup in the gap can lowergap height, also lowering microphone sensitivity.

Signal-to-noise ratio (SNR) is the main competitive performance issue inthe commercial microphone market, which encompasses microphones fordevices such as smartphones, in-ear headphones and hearing aides.Typically, the SNR of commercial MEMS microphones ranges between 55 and65 dB for a sensor area of approximately 1 mm². In microphones, SNR ismeasured when the input acoustic signal level is 94 dBA. The unit dBArefers to A-weighted decibels, which accounts for the human ear'sdifferent perception of loudness at different frequencies.

SNR is defined as the ratio of: the root-mean-square (rms) voltageacross the terminals of the microphone, when the microphone is placed ona rigid baffle and a free field pressure wave of 1 Pa rms amplitude at 1kHz frequency is incident on the microphone; to the rms voltage acrossthe terminals of the microphone, filtered using A-weighted filters, whenthe microphone is completely isolated from any sound sources, such as inan anechoic chamber. The sound level at 0 dBA, which corresponds toabout 20 μPa rms, is accepted as the hearing threshold of the human ear(though clinically measured threshold levels are much louder). Themaximum possible SNR is about 94 dB, because the inherent noise inducedby acoustic radiation physics (the radiation resistance, describedbelow, which provides a generally-applicable noise floor) is about 0 dBAin a microphone with 1 mm² area.

A rigid baffle is an infinite, perfectly reflecting surface around theboundary of an acoustic aperture of a microphone. If a microphone ismounted on a rigid baffle, the incoming acoustic wave will create twicethe free field pressure on the microphone's vibrating element that itwould in empty space.

Noise in a microphone, which reduces the maximum possible SNR of themicrophone, predominantly comes from one of three sources: radiationresistance of the membrane; mechanical losses caused by molecularfriction in the material of vibrating parts, and/or by macroscopicfriction of mechanical parts in the microphone moving against eachother; and in pressure compensated microphones, mechanical losses causedby fluid friction, including the friction of air moving throughperforations (holes) in a membrane or substrate, and the squeezed filmfriction effect in the gap. There can be other losses, such aselectrical energy loss from dielectric loss in the insulator layer. Somepressure compensated MEMS microphones have a noise floor of about 30dBA, with pressure compensation contributing most of this noise. Thenoise floors in pressure compensated electret microphones are generallyhigher than in comparable MEMS microphones.

Radiation resistance is the real component of radiation impedance (acomplex number). Radiation impedance relates to Newton's third law ofmotion: every action has a reaction of equal magnitude and in theopposite direction. A transmitting acoustic transducer (such as aloudspeaker) applies a force onto the medium (pushes the medium, such asair, to and fro) at its aperture during transmission. The medium alsoexerts a reaction force on the transducer surface. The reaction force isequal to the product of the velocity of the transducer surface (theaperture) and the radiation impedance. Radiation impedance is a complexnumber with two components: radiation resistance (the real component)and radiation reactance (the imaginary component). Part of the reactionforce, corresponding to the radiation resistance, generates acousticwaves, which radiate out from the aperture into the medium. The energycomprising the radiated acoustic waves (corresponding to the radiationresistance) is lost with respect to the transducer (the transducer doesnot recover the energy used to create the acoustic waves).

Acoustic transmission and acoustic reception are reciprocal phenomena.Therefore, radiation impedance is also present in acoustic reception(microphones). Radiation resistance is a source of noise in acousticreception. The noise generated by radiation resistance is the noisefloor of a 100% efficient microphone with no other sources of mechanicalor electrical energy loss.

When an acoustic wave is incident on the microphone membrane, theacoustic field energy is included in the transduction and a force isapplied on the membrane surface, which moves the membrane. The reactionforce of the membrane, applied onto the medium (the ambient), is equalto the product of the radiation impedance and the velocity of themembrane. The incident acoustic energy is first partly dissipated by theresistive part of the radiation impedance. Remaining energy is thenavailable to the transduction mechanism (that is, acoustic reception ina microphone). Radiation resistance is an energy dissipative factor intransduction, and therefore generates noise during reception.

The squeeze film effect refers to two consequences of air periodicallysqueezed between a vibrating membrane and a static substrate: (1)increasing air pressure forces air to escape from the gap throughavailable outlets, e.g. holes, causing friction, which dissipates(loses) energy; and (2) increasing air pressure in the gap increases thetemperature of the temporarily compressed (squeezed) air (followingGay-Lussac's Law), which causes energy loss by converting mechanicalenergy into heat.

Some typical integrated commercial MEMS microphones used in mobilephones are operated with a dc bias voltage of 10-14 volts, with anapproximately 28-30 dBA noise floor in their audio bandwidth. Thisamount of self noise corresponds to an SNR of 66 dB or less at thetransducer output before pre-amplification, when the incident signallevel is 1 Pa. Such commercial MEMS microphones typically have about −38dB re V/Pa maximum OCRV (open circuit receive voltage) sensitivity.

A Capacitive Micromachined Ultrasonic Transducer (CMUT) is a capacitivetransducer. CMUTs can be used to transmit and receive ultrasonics. CMUTshave a wide bandwidth in water and in a frequency range near their first(lowest) resonance frequency. Microphones generally have manyresonances. At a resonance, the amount of applied force, externalpressure or electromechanical force required to induce high-amplitudevibration of the membrane is reduced. Ultrasonic transducers (such asCMUTs) are usually operated near their first resonance frequency. Thisenables the transducers to be highly sensitive; however, for efficienttransmission and/or reception to be maintained, the transducer will haveeither a narrow operation bandwidth, or increased internal loss andconsequent increased noise (lower SNR). Internal loss is power loss, andis the sum of power lost through mechanical and electrical energy lossmechanisms other than radiation resistance.

In some examples, CMUTs can have a pressure compensated gap, resultingin a compliant radiation plate and a relatively wide bandwidth. In someexamples, CMUTs can have a sealed gap, resulting in low internal loss(in some examples, less than their radiation resistance in air). CMUTsare typically characterized as receivers when operated at a resonancefrequency, and as microphones when operated off-resonance. A sealed gapcan contain a sealed-in gas, or a vacuum (a “vacuum gap”). Internal lossin CMUT transducers is typically small with respect to the noiseintroduced by radiation resistance—small enough to be difficult toaccurately measure. In some examples, losses and radiation impedance insealed gap airborne CMUTs generate about 0 dBA in the audio bandwidth,which is slightly more than the noise contribution of the CMUT'sradiation resistance in a 1 mm² microphone operated off-resonance in anaudible range (generally, about 10 Hz to 20 kHz).

A pressure compensated MEMS microphone comprising a transducer, sealedmembranes and a sealed volume is disclosed by U.S. Pat. No. 6,075,867.

An integrated and programmable microphone bias generation system isdescribed by U.S. Pat. No. 8,288,971.

An implantable microphone which uses a housing to hermetically seal themicrophone is described in U.S. Pat. No. 9,451,375. This microphonecompensates for noise artifacts caused by the housing by using twohighly compliant parallel membranes, compliance of the membranes beingenhanced by respective pressure compensated gaps.

An implantable microphone which uses a perforated membrane for pressurecompensation is described in U.S. Pat. No. 7,955,250. The perforation inthe membrane makes the membrane more compliant, and thus increasessensitivity. U.S. Pat. No. 9,560,430 also describes a microphone with aperforated membrane.

A microphone module which uses vents to enable pressure compensation,and for driving water out of the system, is described by U.S. Pat. Pub.No. 2015/0163572.

A pressure compensated microphone module for a phone watch that uses ahydrophobic plate covered by an “impermeable” membrane—which allowspassage of gasses—to enable pressure compensation, and to keep water outof the microphone, is described by Pat. Pub. No. 2001/0019945.

Some microphones use hydrophobic and/or oleophobic materials to covermicrophone components to protect them from fluids. For example, amicroporous composite material containing polytetrafluoroethylene (PTFE)is described in Pat. Pub. No. 2014/0083296 for use in filters, vents orprotective membranes. PTFE is gas permeable such that it can both beused as a protective membrane and enable pressure compensation. Ahydrophobic mesh (umbrella-shaped, covering an acoustic port), isdescribed in U.S. Pat. No. 9,363,589. However, PTFE, hydrophobic mesh,and other methods of “waterproofing” microphones with pressurecompensated gaps will generally degrade performance (due to isolation ofsound-detection membranes from sound sources), and will fail to protecttransducers from water given a relatively small static pressuredifference between the external environment (e g , immersion in water ata depth of a meter) and the gap, or given repeated submersion.

A MEMS microphone with a piezoelectric (rather than capacitive orelectret) membrane, which can be covered by a Parylene film forwaterproofing, is described in U.S. Pat. Pub. 2014/0339657.Piezoelectric MEMS microphones are fabricated using different productionprocesses than capacitive microphones.

The inventors endeavor to disclose new and advantageous approaches to acapacitive MEMS microphone with a sealed gap, and methods for designingsuch microphones, as further described below.

SUMMARY

Some preferred embodiments include a microphone system for receivingsound waves, the microphone including a back plate, a radiation plate,first and second electrodes, first and second insulator layers, a powersource and a microphone controller. The radiation plate is clamped tothe back plate so that there is a hermetically sealed circular gapbetween the radiation plate and the back plate. The first electrode isfixedly attached to a side of the back plate proximate to the gap. Thesecond electrode is fixedly attached to a side of the radiation plate.The insulator layers are attached to the back plate and/or the radiationplate, on respective gap sides thereof, so that the insulator layers arebetween the electrodes. The microphone controller is configured to usethe power source to drive the microphone at a selected operating pointcomprising normalized static mechanical force, bias voltage, andrelative bias voltage level. A radius and height of the gap, and athickness of the radiation plate, are determined using the selectedoperating point so that a sensitivity of the microphone at the selectedoperating point is an optimum sensitivity for the selected operatingpoint.

Numerous other inventive aspects are also disclosed and claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosed inventive subject matter will be described with referenceto the accompanying drawings, which show important sample embodimentsand which are incorporated in the specification hereof by reference,wherein:

FIG. 1 schematically shows a cross-section of an example of a pressurecompensated MEMS microphone.

FIG. 2 schematically shows a cross-section of an example of a pressurecompensated electret microphone.

FIG. 3 schematically shows an example of a cross section view of aMicromachined Capacitive Microphone (MCM) with an undeflected radiationplate.

FIG. 4 schematically shows an example of a cross section view of a MCMwith a depressed radiation plate.

FIG. 5 shows a graph of the relationship between the ratio of the biasvoltage to the collapse voltage in a vacuum V_(DC)/V_(r) and thenormalized static displacement of the center of the radiation plate 310X_(P)/t_(ge) at the electromechanical equilibrium (the equilibriumpoint).

FIG. 6 shows a graph of the relationship between normalized effectivegap height t_(ge) _(_) _(n) and normalized static mechanical forceF_(b)/F_(g) for an MCM.

FIG. 7 shows a lin-log semi-log graph of the relationship between themaximum normalized radiation plate radius-to-thickness ratio(a/t_(m))_(N) _(_) _(max) enables an MCM to meet the elastic linearityconstraint, and normalized static mechanical force F_(b)/F_(g), forexample values of the relative bias voltage level V_(DC)/V_(C).

FIG. 8A shows a log-lin semi-log graph of the relationship betweennormalized minimum gap radius a_(n) _(_) _(min) that enables an MCM tomeet the elastic linearity constraint, and normalized static mechanicalforce F_(b)/F_(g), for example values of the relative bias voltage levelV_(DC)/V_(C).

FIG. 8B shows a log-lin semi-log graph of the relationship betweennormalized minimum radiation plate thickness t_(m) _(_) _(n) _(_) _(min)that enables an MCM to meet the elastic linearity constraint, andnormalized static mechanical force F_(b)/F_(g), for example values ofthe relative bias voltage level V_(DC)/V_(C).

FIG. 9 shows a semi-log graph of the relationship between normalizedOpen Circuit Received Voltage Sensitivity (OCRV) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C), where parasitic capacitance C_(P) divided byclamped capacitance C₀ equals zero (C_(P)/C₀=0).

FIG. 10 shows a log-lin semi-log graph of the relationship betweennormalized input capacitance C_(in) _(_) _(n) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C), where the normalized radius-to-thicknessratio equals the maximum normalized radius-to-thickness ratio(a/t_(m))_(N) which enables linearly elastic operation.

FIG. 11 shows a graph of the relationship between normalized ShortCircuit Received Current Sensitivity (SCRC) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C).

FIG. 12 shows a graph of the relationship between normalized ShortCircuit Received Current Sensitivity (SCRC) per square meter andnormalized static mechanical force F_(b)/F_(g), for example values ofthe relative bias voltage level V_(DC)/V_(C).

FIG. 13 shows an example process for design of an MCM, starting with aselected OCRV sensitivity, normalized static mechanical forceF_(b)/F_(g), and relative bias voltage V_(DC)/V_(C).

FIG. 14 shows an example process for design of an MCM, starting with aselected normalized static mechanical force F_(b)/F_(g), bias voltageV_(DC), and relative bias voltage V_(DC)/V_(C).

FIG. 15 shows an example process for design of an MCM, starting with aselected gap radius a, bias voltage V_(DC), and radiation plate 310material.

FIG. 16 shows an example process for design of an MCM, starting with aselected radiation plate thickness t_(m) or effective gap height t_(ge),and a selected bias voltage V_(DC) or other parameter dependent on biasvoltage V_(DC).

FIG. 17 shows a graph of the relationship between normalized minimum gapradius a_(n) _(_) _(min) and normalized effective gap height t_(ge) _(_)_(n) for various values of the relative bias voltage level V_(DC)/V_(C).

FIG. 18 shows a graph of the relationship between normalized minimumradiation plate thickness t_(m) _(_) _(n) _(_) _(min) and normalizedeffective gap height t_(ge) _(_) _(n) for various values of the relativebias voltage level V_(DC)/V_(C).

FIG. 19 shows a graph of the relationship between normalized gap radiusa_(n) and normalized effective gap height t_(ge) _(_) _(n) for variousvalues of the relative bias voltage level V_(DC)/V_(C) and variousvalues of the scaling constant K.

FIG. 20 shows a graph of the relationship between normalized radiationplate thickness t_(m) _(_) _(n) and normalized effective gap heightt_(ge) _(_) _(n) for various values of the relative bias voltage levelV_(DC)/V_(C) and various values of the scaling constant K.

DETAILED DESCRIPTION OF SAMPLE EMBODIMENTS

The numerous innovative teachings of the present application will bedescribed with particular reference to presently preferred embodimentsby way of example, and not of limitation. The present applicationdescribes inventive scope, and none of the statements below should betaken as limiting the claims generally.

The present application discloses new approaches to capacitive MEMSmicrophones with a sealed gap, and to design of such microphones.

Some exemplary parameters will be given to illustrate the relationsbetween these and other parameters. However it will be understood by aperson of ordinary skill in the art that these values are merelyillustrative, and will be modified by scaling of further devicegenerations, and will be further modified to adapt to differentmaterials or architectures if used.

A capacitive MEMS microphone with a sealed gap is disclosed herein whichis preferably an airborne microphone configured for off-resonanceoperation (described below with respect to FIG. 3). Such microphones arereferred to herein as Micromachined Capacitive Microphones (MCM).

The inventors have made the surprising discovery that MCMs can beconstructed with gap and vibrating membrane dimensions that result inrobust uncollapsed, linearly elastic operation with high sensitivity andlittle or no self-noise—in some embodiments, an SNR of approximately 94dBA can be achieved across the audible spectrum! Further, because MCMsare sealed, they are waterproof, in some embodiments down to tens ofmeters in depth.

The inventors have also made the surprising discovery that certain MCMoperating parameters and MCM gap and vibrating membrane dimensions aredeterministically related, such that MCM dimensions which will result inhigh sensitivity (or optimal sensitivity for selected operatingparameters) can be determined from selected operating parameters. Inother words, microphone design can be performed backwards for MCMs,starting from selected performance requirements, which can be used todetermine corresponding physical microphone dimensions which will resultin those performance characteristics! Moreover, if an MCM microphone ismade from solid materials suitable for MEMS device fabrication, thedetermined dimensions will generally be unaffected by the particularmaterials used!

MCMs are related to CMUTs, but preferably operate in an audible range.MCMs can be used in, for example, airborne consumer and professionalproducts, such as computers, ear phones, hearing aids, mobile phones,wireless equipment and wideband precision acoustic measurement andrecording systems. Preferred MCM embodiments comprise a relativelysimple structure which can be fabricated at low cost using standard MEMSprocesses.

In an MCM, dimensions of the microphone that optimize microphonesensitivity, SNR and other performance characteristics can be determinedby selecting values for three operating parameters (an “operatingpoint”): normalized static mechanical force F_(b)/F_(g), bias voltage ofelectrodes V_(DC), and relative bias voltage V_(DC)/V_(C). (When notspecified, “sensitivity” herein refers to the Open Circuit ReceiveVoltage (OCRV) sensitivity.) The operating point, including the collapsevoltage V_(C), is further described below, along with the relationshipsbetween the operating point, MCM dimensions, MCM sensitivity and otherMCM parameters. Further, the operating point can be used to determinenormalized values for microphone dimensions which are independent ofproperties of materials used in fabricating the microphone.De-normalized microphone dimensions (physical dimensions forfabrication) can then be determined from normalized dimensions usingelastic properties (Young's modulus and Poisson's ratio) of a vibratingelement (radiation plate), a static differential pressure between thegap and the ambient atmosphere (referred to herein as the ambient), andthe permittivities of insulator layers connected to gap-facing sides ofthe radiation plate. These relationships are described below.

A model relating various dimensions and properties of CMUTs is developedin H. Köymen, A. Atalar, E. Aydoğdu, C. Kocabas̨, H. K. Oğuz, S. Olçum,A. Özgürlük, A. Ünlügedik, “An improved lumped element nonlinear circuitmodel for a circular CMUT cell,” IEEE Trans. Ultrason. Ferroelectr.Freq. Control, Vol. 59, no. 8, pp. 1791-1799, August 2012, which isincorporated herein by reference (and referred to herein as the “CircuitModel reference”). This model is further developed in H. Köymen, A.Atalar and H. K. Oğuz, “Designing Circular CMUT Cells Using CMUT BiasingChart,” 2012 IEEE International Ultrasonics Symposium Proceedings pp.975-978, Dresden, October, 2012 (the “CMUT Design reference”). As MCMstructure is based on principles of CMUT operation, the model developedin the Circuit Model and CMUT Design references is relevant to MCMdesign. However, the relationships described herein enablingdetermination of MCM measurements and OCRV sensitivity from an operatingpoint were not stated in the Circuit Model and CMUT Design references.

A single capacitive microphone, such as an MCM, is also called a “cell”.A microphone system can comprise multiple cells.

FIG. 3 schematically shows an example of a cross section of aMicromachined Capacitive Microphone 300 (MCM), comprising a capacitiveelectroacoustic microphone with a sealed gap 302. As shown in FIG. 3, anMCM 300 preferably comprises a circular gap 302 fabricated (e.g.,machined or etched) into a surface of a substrate 304, with thesubstrate 304 at the bottom of the gap 302 forming a back plate 306.(Non-circular gaps are described with respect to and following Equations30-34, below.) The back plate 306 is made of a solid material suitablefor use in manufacturing MEMS microphones, such as a metal, aconducting, semiconducting or insulating ceramic, or a crystalline orpolycrystalline material. A bottom electrode 308 is formed over the backplate 306, e.g., using a metallization technique.

A vibrating element in a microphone that is used to measure acousticenergy is generally called a “membrane” or a “radiation plate” dependingon the vibrating element's radius-to-thickness ratio. If the vibratingelement's radius-to-thickness ratio is less than a threshold (whichdifferent authorities specify as, for example, 40, 80 or 100), then thevibrating element is a “radiation plate”; otherwise, it is a “membrane”.MCMs 300 will generally use a vibrating element with aradius-to-thickness ratio less than 40. (This is discussed below withrespect to FIG. 7, using the scaling constant term first described withrespect to Equation 14.) Therefore, the vibrating element in MCMsdescribed herein is referred to as a “radiation plate”.

A radiation plate 310 of total thickness t_(m) (thickness of membrane)is clamped to the back plate 306 at the aperture of the gap 302 (theupper side of the gap 302, that is, the side distant from the back plate306), preferably at the rim of the gap's 302 aperture, such that the gap302 is sealed. (“Total” thickness refers to t_(m) being the sum of thethickness of the radiation plate 310, plus any electrodes or insulatorlayers, further described below, which are attached to it.) To implementthis clamping and seal, the substrate 304 and the radiation plate 310are mechanically coupled, e.g., by bonding, wafer bonding or sacrificiallayer processing. The gap 302 is preferably completely (hermetically)sealed, so that no air (or other gas, dust or other material) can passbetween the gap 302 and the ambient. The radiation plate 310 can be madeof a solid material generally suitable for MEMS manufacture, such as ametal, a conducting, semiconducting or insulating ceramic, or acrystalline or polycrystalline material.

The radiation plate 310 can comprise multiple layers of differentmaterials, such as a metal layer (or layers) for an electrode, a layerfor compliance (C_(RM)), and an insulator layer. The elastic propertiesof one layer will generally be more significant than the elasticproperties of the other layers, since the other layers will generally becomparatively thin. The combined effects of multilayer structures onelastic behavior of a vibrating element in a microphone are describedby: M. Funding la Cour, T. L. Christiansen, J. A. Jensen, Fellow, IEEE,and E. V. Thomsen, “Electrostatic and Small-Signal Analysis of CMUTsWith Circular and Square Anisotropic Plates,” IEEE Trans. Ultrason.Ferroelectr. Freq. Control, vol. 62, no. 8, pp. 1563-1579, 2015 (the“Anisotropic Plates” reference). This reference provides an approach totreating a multilayered vibrating element as an equivalent single layervibrating element, and determining a Young's modulus and Poisson's ratiofor the equivalent single layer vibrating element.

Airborne MCMs 300 (MCMs operated in air) are preferably operatedoff-resonance. This is because an MCM 300 operated on-resonance wouldhave a high sensitivity peak, but the bandwidth would be relativelynarrow (in some embodiments, too narrow for typical consumer electronicsimplementations such as cellular phone microphones).

The gap 302 has a radius a and a gap height t_(g). (The gap radius a isalso the radius of the radiation plate 310.) The gap height t_(g) is thedistance between the uppermost material at the bottom of the gap 302 andthe lowermost material at the top of the gap 302 when the radiationplate 310 is undeflected. The radiation plate 310 is undeflected whenthe normalized static mechanical force F_(b)/F_(g) equals zero(generally, when there is no static pressure difference between the gap302 and the ambient), and the bias voltage V_(DC) is zero or therelative bias level V_(DC)/V_(C) equals zero. F_(b)/F_(g), V_(DC)/V_(C)and the “collapse voltage” V_(C) are further described below. A smallergap 302 radius a or a larger radiation plate 310 thickness t_(m) willincrease the stiffness of the radiation plate 310.

A top electrode 312 is fixedly connected to the radiation plate 310, orcan be the radiation plate 310 itself if the radiation plate 310 is madeof a conductive material. The top electrode 312 can be formed on eithersurface of the radiation plate 310, or can be formed within theradiation plate 310 if the radiation plate 310 is made of a dielectricmaterial. The top electrode 312 is preferably formed using ametallization technique (if the radiation plate 310 is not itself thetop electrode 312). Preferably, the bottom electrode 308 fully coversthe back plate 306 (the bottom of the gap 302; that is, the back plate306 is “fully electroded”), and the top electrode 312 fully covers theportion of the radiation plate 310 that faces and touches the gap 302(the radiation plate 310 is “fully electroded”). The voltage across theelectrodes 308, 312 is a bias voltage V_(DC). Generally, at lower biasvoltages V_(DC), better microphone performance is achieved if the backplate 306 and radiation plate 310 are fully electroded. Electrodes 308,312 can also be smaller than the gap 302, down to 80% of the size of thegap 302, as further explained below. Electrodes 308, 312 which aresmaller than the gap 302 are preferably concentric with the gap 302.

There is preferably a first dielectric insulator layer 314 of thicknesst_(i1) fixedly attached to and covering the gap 302 side of the bottomelectrode 308, and a second dielectric insulator layer 316 of thicknesst_(i2) fixedly attached to and covering the gap 302 side of thecombination of the radiation plate 310 and the top electrode 312. Inalternative embodiments, both of the dielectric insulator layers 314,316 can be located on the gap 302 side of either the bottom electrode308, or the combination of the radiation plate 310 and the top electrode312. The insulating layers 314, 316 can be made of an insulatingmaterial suitable for use in a MEMS microphone (generally, any suchmaterial), such as an insulating ceramic, polymer, crystalline orpolycrystalline material. One or both of the insulator layers 314, 316can be electrets.

Electrets and certain CMUT performance measurements are addressed in H.Köymen, A. Atalar, Itir Köymen, A. S. Tas̨delen, A. Ünlügedik, “UnbiasedCharged Circular CMUT Microphone: Lumped Element Modeling andPerformance”, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 65,no. 1, pp. 60-71, Nov. 14, 2018, which is incorporated herein byreference (and referred to herein as the “Electret and Performancereference”). The Electret and Performance reference and the AnisotropicPlates reference show that noise (losses) in a CMUT (a capacitive MEMSmicrophone with a sealed gap) are very small—in some embodiments,approximately 0 dBA.

An MCM 300 is a capacitive microphone. Capacitive microphone operationuses the fact that if a voltage (electric potential) is applied acrosstwo parallel conducting plates (the bottom and top electrodes 308, 312)separated by a gap 302, the parallel conducting plates 308, 312 willattract each other electrostatically via the electromechanicalattraction force. The radiation plate 310 is clamped (fixedly connected)to the substrate 304 at the rim of the gap 302, and the top electrode312 is attached to (fixedly connected to or comprised of) the radiationplate 310. Because the radiation plate 310 is clamped to the substrate304 at the rim of the gap 302, the spring reaction (elastic restoringforce) due to the elasticity of the radiation plate 310 resists theelectromechanical force exerted by the top electrode 312. That is, theattraction between the electrodes 308, 312 pulls the radiation plate 310down into the gap 302, and the elasticity of the radiation plate 310pulls the radiation plate 310 back towards a resting position. Thevoltage across the electrodes 308, 312 is the bias voltage V_(DC). For agiven bias voltage V_(DC), the electromechanical force and elasticrestoring force are balanced when the center of the radiation plate 310is displaced by an equilibrium displacement distance (also called theequilibrium point).

As stated, the voltage across the electrodes 308, 312 is a bias voltageV_(DC). If the bias voltage V_(DC) is increased beyond a limit for“uncollapsed” microphone operation called the “collapse voltage” V_(C),the elastic restoring force is unable to prevent the electromechanicalforce from causing the center of the radiation plate 310 to collapseinto (make physical contact with) the bottom of the gap 302. In exampleembodiments as shown in FIG. 3, this would comprise the first insulatorlayer 314 touching the second insulator layer 316. Generally, microphoneSNR is significantly decreased in collapsed operation. The ratio betweenthe bias voltage V_(DC) and the collapse voltage V_(C) is called therelative bias level V_(DC)/V_(C).

Preferably, the sealed gap 302 contains a very low pressure environment(a vacuum, for example, less than 10 mbar). If the gap 302 contains avacuum, there is a static pressure difference P₀ between the ambientenvironment (on the other side of the radiation plate 310 from the gap302) and the gap 302 which results in a net static force F_(b) pushingthe radiation plate 310 into the gap 302.

At equilibrium, when sound (a time varying pressure signal) is incidenton the radiation plate 310 (accordingly, received by the MCM 300), theradiation plate 310 vibrates and the displacement of the radiation plate310 changes (e.g., oscillates) around the equilibrium point. Thismovement causes variation of the microphone capacitance (the capacitancebetween the top and bottom electrodes 308, 312). Variation in themicrophone capacitance, combined with the charge stored on thecapacitance due to the bias voltage V_(DC), causes a voltage across theoutput terminals of the microphone to vary in proportion to the incidentsound pressure signal. This output voltage can be amplified, measured,stored, and used to reproduce (play back) the sound originally receivedby the microphone (the MCM 300).

An “operating point” is defined herein as a triplet of selected valuescomprising the applied bias voltage V_(DC), the relative bias levelV_(DC)/V_(C), and the normalized static mechanical force F_(b)/F_(g)(further described below with respect to FIG. 4 and Equation 6). Asdisclosed below, an operating point uniquely determines dimensions of anMCM 300 that will result in optimal sensitivity of the MCM 300 at thatoperating point. For example, an operating point can be used todetermine an MCMs 300 gap 302 radius a, radiation plate 310 thicknesst_(m), and effective gap 302 height t_(ge) (as described below withrespect to, for example, Equations 9-18). Alternatively, the operatingpoint can be used to determine an MCM's minimum gap 302 radius a_(min),minimum radiation plate 310 thickness t_(m) _(_) _(min), and effectivegap 302 height t_(ge), along with a range for radiation plate 310radius-to-thickness ratio a/t_(m) enabling the MCM 300 to maintainelastic linear operation (operation within the elastic linearityconstraint, as described below with respect to FIGS. 7, 8A and 8B).Dimensions as determined yield a resulting (and optimal) open circuitreceived voltage (OCRV) sensitivity at a corresponding operating point.Dimensions can then be adjusted to enable robust elastic linearoperation without compromising the OCRV sensitivity. Various examples ofMCM 300 design processes are shown with respect to, for example, FIGS.13-16. These results take advantage of the very low noise floor (in someembodiments, approximately 0 dBA) and high SNR (in some embodiments,approximately 94 dBA) in airborne sealed gap 302 MCMs 300 as disclosedherein.

The operating point can be selected: for example, to minimize biasvoltage V_(DC), and/or to correspond to a selected OCRV sensitivity, gap302 radius a (or other physical dimension), or other desired performancecharacteristic. Selectable operating point values, and optimality ofresults with respect to the selected operating point, are not limited bymaterials to be used in fabrication of the radiation plate 310 orinsulator layers 314, 316. Such components in an MCM 300 can be made outof materials suitable for manufacture of similar components in MEMSdevices (in preferred embodiments, any such materials). Normalizeddimensions of the MCM 300, which are not dependent on materialproperties, can be determined directly from the operating point.De-normalized dimensions used before MCM 300 fabrication can then bedetermined using properties of materials selected for use in MCM 300components. As a result, dimensions, sensitivity and other properties ofthe MCM, including gap 302 radius a and radiation plate 310 thicknesst_(m), effective gap 302 height t_(ge), and Open Circuit ReceivedVoltage Sensitivity (OCRV), as well as other microphone performanceparameters, are independent of the particular material(s) used tofabricate the radiation plate 310 and the insulator layers 314, 316.

Also described herein are conditions enabling the gap 302 radius a, theradiation plate 310 thickness t_(m), and the ratio between the gap 302radius and the radiation plate 310 thickness a/t_(m) to be rescaled,within ranges and with relationships determined by the operating point,while maintaining the optimal OCRV sensitivity for that operating point.

FIG. 4 shows an example visual representation 400 of an analytical modelfor a Micromachined Capacitive Microphone 300 (MCM), using across-section of the MCM 300. As shown in FIG. 4, the effective gap 302height t_(ge), which is an electrical dimension of the gap 302 used inmodeling the MCM 300, depends on the gap height t_(g), the relativepermittivity of the first insulator layer 114 ε_(r) _(_) _(i1), and therelative permittivity of the second insulator layer 116 ε_(r) _(_)_(i2). These relative permittivities are the ratios between therespective permittivities of the insulator layers 314, 316 and thepermittivity of free space. (Permittivity is the resistance of a mediumto forming an electric field in that medium. The gap 302 preferablycontains a vacuum, which has a relative permittivity of 1.) Theeffective gap height t_(ge) is determined as shown in Equation 1. Notethat if the entire gap 302 height t_(g) and insulator height (t_(il)plus t_(i2)) comprised vacuum, the effective gap height t_(ge) wouldequal the gap height t_(g).

$\begin{matrix}{t_{ge} = {t_{g} + \frac{t_{i\; 1}}{ɛ_{{r\_ i}\; 1}} + \frac{t_{i\; 2}}{ɛ_{{r\_ i}\; 2}}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

Insulator layer 314, 316 thicknesses and materials (corresponding topermittivities) can be selected after the effective gap 302 heightt_(ge) is determined. That is, appropriate materials for insulator layer314, 316 fabrication can be selected to keep insulator layer 314, 316thickness (t_(i1), t_(i2)) small relative to the gap 302 height t_(g).Once effective gap 302 height t_(ge) is determined, then gap 302 heightt_(g) can be determined such that gap 302 height t_(g) is greater thanthe static displacement of the center of the radiation plate 310 X_(P),plus a margin for production tolerances and insulator layer 314, 316thicknesses using selected insulator materials. The static displacementof the center of the radiation plate 310 X_(P) is the deflectiondistance of the center of the radiation plate 310 from the effective gapheight t_(ge) at the equilibrium point. Higher relative permittivitiesof insulator layers 314, 316 generally correspond to thinner insulatorlayers 314, 316. The effective gap 302 height t_(ge) is determined fromthe operating point as shown below in Equations 8 through 12.

Microphones are more sensitive when the bias voltage V_(DC) is larger.The effective gap 302 height t_(ge) determines the level of bias voltageV_(DC) that can be used, because higher bias voltages increase thedeflection of the radiation plate 310, and sufficiently high biasvoltages V_(DC) will cause the radiation plate 310 to collapse. Voltageavailable on a device also limits bias voltage V_(DC). For example, somemobile phones are limited to about 14 volts available to mobile phonecomponents. Electrets can provide, for example, 150 volts to 200 voltsbias voltage. The Electret and Performance reference is relevant toimplementation of electrets in a capacitive MEMS microphone with asealed gap.

In an MCM 100, the bias voltage V_(DC), the static displacement of thecenter of the radiation plate 310 X_(P), and the net static force on theradiation plate 310 due to the ambient static pressure F_(b) arerelated, in static electromechanical equilibrium (at the equilibriumpoint), as shown in Equation 7 (below).

The relationship shown in Equation 7 is dependent on various propertiesof the MCM 300 (which are explained below), including the shape functionof a deflected clamped circular plate g(X_(P)/t_(ge)) (also referred toas g(u)), which is proportional to the capacitance of the MCM 300; thetransduction force (proportional to g′(u), the first derivative ofg(u)); the collapse voltage in vacuum V_(r) (a reference voltage); thenormalized static mechanical force F_(b)/F_(g); the Young's modulus Y₀(stiffness) and Poisson's ratio σ (signed ratio of transverse strain toaxial strain) of the radiation plate 310; the differential pressure P₀between the ambient static pressure and the pressure in the gap 302; theclamped capacitance C₀, and the compliance of the radiation plate 310C_(Rm) (the inverse of the stiffness of the radiation plate 310).

The transduction force is the force generated on the radiation plate 310when a bias voltage V_(DC) is applied. Equation 3 expresses thetransduction force in terms of the effect the bias voltage V_(DC) has onthe shape of the radiation plate 310 (rather than in terms of the biasvoltage V_(DC)). The variable u corresponds to the ratio of the staticdisplacement to the effective gap height X_(P)/t_(ge).

$\begin{matrix}{{g(u)} = \frac{\tanh^{- 1}\left( \sqrt{u} \right)}{\sqrt{u}}} & {{Equation}\mspace{14mu} 2} \\{{g^{\prime}(u)} = {\frac{1}{2\; u}\left( {\frac{1}{1 - u} - {g(u)}} \right)}} & {{Equation}\mspace{14mu} 3} \\{{g^{''}(u)} = {\frac{1}{2\; u}\left( {\frac{1}{\left( {1 - u} \right)^{2}} - {3\; {g^{\prime}(u)}}} \right)}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

Equation 5 shows the collapse voltage in vacuum V_(r) for a fullyelectroded MCM 300. V_(r) depends on dimensions of the MCM 300 andproperties of the radiation plate 310. This model is also valid for MCMs300 using electrodes 308, 312 which are between 80% and 100% of the sizeof the gap 302 area, if g(u) and its derivatives (that is, the termsused to determine the transduction force and the shape function of theradiation plate 310) are modified as shown in the Circuit Modelreference.

$\begin{matrix}{V_{r} = {\sqrt{\frac{4\; t_{ge}^{2}}{15\; C_{Rm}C_{0}}} = {8\; t_{ge}\frac{t_{m}^{2}}{a^{2}}\sqrt{\frac{t_{ge}}{t_{m}}}\sqrt{\frac{Y_{0}}{27{ɛ_{0}\left( {1 - \sigma^{2}} \right)}}}}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

As previously stated, P₀ is the differential pressure between theambient static pressure and the pressure in the gap 302. For example, ifthe gap 302 contains a vacuum and the ambient static pressure equalsStandard Atmospheric Pressure (SAP), then P₀ equals SAP.

As previously stated, F_(b) is the net static force on the radiationplate 310 due to the ambient static pressure, that is, the force on theradiation plate 310 due to the differential static pressure between theambient static pressure and the pressure in the gap 302 P₀. F_(g) is theuniformly distributed force required to displace the center of theradiation plate 310 by the effective gap height t_(ge) (that is, tocause the radiation plate 310 to collapse). Because t_(ge)≥t_(g) inuncollapsed operation (depending on whether there is an insulator layer314, 316 between the electrodes 308, 312, see Equation 1), thenormalized static mechanical force F_(b)/F_(g)≤1. The normalized staticmechanical force F_(b)/F_(g) is given in Equation 6.

$\begin{matrix}{\frac{F_{b}}{F_{g}} = {\frac{\pi \; a^{2}{P_{0}/3}}{{t_{ge}/5}\; C_{Rm}} = {\frac{3\left( {1 - \sigma^{2}} \right)P_{0}}{16\; Y_{0}}\frac{a^{4}}{t_{m}^{4}}\frac{t_{m}}{t_{ge}}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

In an MCM 300 in uncollapsed operation in which the gap 302 contains avacuum, the normalized static mechanical force F_(b)/F_(g) can assumevalues between 0 (if the ambient static pressure is zero, so thatdifferential static pressure P₀=0; or if the radiation plate 310 isinfinitely stiff, meaning

$\left. \frac{1}{C_{RM}}\rightarrow\infty \right)$

and the ratio between the gap 302 height and the effective gap 302height t_(g)/t_(ge). The limiting case F_(b)/F_(g)=1 means that thecenter of the radiation plate 310 is displaced by the effective gap 302height t_(ge), which is not physically possible when there is aninsulator layer 314, 316 between the electrodes 308, 312. (F_(b)/F_(g)is also zero in pressure compensated MEMS microphones.)

The normalized static mechanical force F_(b)/F_(g) will generally berelatively low in an MCM 300 with a stiff radiation plate 310

$\left( {{large}\frac{1}{C_{RM}}} \right),$

or with a compliant radiation plate 310 (large C_(Rm)) and a largeeffective gap 302 height t_(ge). F_(b)/F_(g) will generally berelatively high if the ambient static pressure displaces the radiationplate 310 by a significant fraction of the effective gap 302 heightt_(ge), which can occur, for example, in a MCM 300 with a compliantradiation plate 310, or with a stiff radiation plate 310 and arelatively small effective gap height t_(ge).

FIG. 5 shows a graph 500 of the relationship between the ratio of thebias voltage to the collapse voltage in a vacuum V_(DC)/V_(r) and thenormalized static displacement of the center of the radiation plate 310X_(P)/t_(ge) at the electromechanical equilibrium (the equilibriumpoint). In FIG. 5, the solid curves correspond to operational domains inwhich the microphone will be in uncollapsed operation; the dotted linemarks the transition between uncollapsed operation and collapsedoperation; and the dotted curves correspond to operational domains inwhich the microphone will be in collapsed operation. The ratio of thebias voltage to the collapse voltage V_(DC)/V_(r) is given in Equation7.

$\begin{matrix}{\frac{V_{DC}}{V_{r}} = {{\sqrt{\frac{3\left( {\frac{X_{P}}{t_{ge}} - \frac{F_{b}}{F_{g}}} \right)}{2\; {g^{\prime}\left( \frac{X_{P}}{t_{ge}} \right)}}}\mspace{14mu} {for}\mspace{14mu} \frac{X_{P}}{t_{ge}}} \geq \frac{F_{b}}{F_{g}}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

Equation 7 shows that the static displacement of the center of theradiation plate 310 X_(P) is equal to t_(ge)*(F_(b)/F_(g)) when theplate is electrically unbiased, so that V_(DC)=0. This can also beviewed as the normalized static displacement of the center of theradiation plate 310 X_(P)/t_(ge) being equal to the normalized staticmechanical force F_(b)/F_(g) when no bias voltage is applied, so thatV_(DC)=0.

The collapse voltage V_(C) depends on the normalized static mechanicalforce F_(b)/F_(g), as well as the stiffness of the radiation plate 310and the effective gap 302 height t_(ge). When the radiation plate 310 isdisplaced by ambient static pressure (accordingly, the MCM 300 is not ina vacuum), the collapse voltage V_(C) is decreased from the collapsevoltage in a vacuum V_(r). As shown in Equation 8, the collapse voltageV_(C), normalized to V_(r), depends only on F_(b)/F_(g).

$\begin{matrix}{\frac{V_{C}}{V_{r}} \approx {0.9961 - {1.0468\frac{F_{b}}{F_{g}}} + {0.06972\left( \frac{F_{b}}{F_{g}} \right)^{2}} + {0.01148\left( \frac{F_{b}}{F_{g}} \right)^{6}}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

As shown in Equations 9 through 18 below, the MCM 300 dimensions, thatis, gap 302 radius a, radiation plate 310 thickness t_(m) and effectivegap 302 height t_(ge), can be expressed in terms of the operating point:normalized static mechanical force F_(b)/F_(g), relative bias levelV_(DC)/V_(C), and bias voltage V_(DC).

The effective gap 302 height t_(ge) is determined as shown in Equation9.

$\begin{matrix}{t_{ge} = {{\frac{3}{2}V_{r}\sqrt{\frac{ɛ_{0}}{P_{0}}\frac{F_{b}}{F_{g}}}} = {\frac{3}{2}\sqrt{\frac{ɛ_{0}}{P_{0}}}{{V_{DC}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}\left\lbrack {\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\sqrt{\frac{F_{b}}{F_{g}}}} \right\rbrack}}}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

Equation 9 can be rewritten to express the effective gap height t_(ge)in terms of the normalized bias voltage V_(DC) _(_) _(n) and thenormalized effective gap 302 height t_(ge) _(_) _(n), as shown inEquation 10. V_(DC) _(_) _(n) is defined as shown in Equation 12.

$\begin{matrix}{t_{ge} = {{V_{DC\_ n}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}t_{ge\_ n}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

The normalized effective gap 302 height t_(ge) _(_) _(n) is a functionof normalized static mechanical force F_(b)/F_(g), as shown in Equation11A.

$\begin{matrix}{{t_{ge\_ n}\left( \frac{F_{b}}{F_{g}} \right)} = {\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\sqrt{\frac{F_{b}}{F_{g}}}}} & {{Equation}\mspace{14mu} 11A}\end{matrix}$

FIG. 6 shows a graph 600 of the relationship between normalizedeffective gap height t_(ge) _(_) _(n) and normalized static mechanicalforce F_(b)/F_(g) for a MCM 300, as described in Equation 11B. Equation11A can be rewritten using Equation 8 so that the normalized effectivegap 302 height t_(ge) _(_) _(n) depends only on normalized staticmechanical force F_(b)/F_(g), as shown in Equation 11B. Equations 10,11B and 12 can be used to determine the effective gap 302 height t_(ge)using the operating point, independent of material properties. Equation1 can be used to determine the gap height t_(g) using the effective gapheight t_(ge) and permittivities of selected insulator layer 314, 316materials.

$\begin{matrix}{{t_{ge\_ n}\left( \frac{F_{b}}{F_{g}} \right)} \approx \frac{\sqrt{\frac{F_{b}}{F_{g}}}}{\begin{matrix}{0.9961 - {1.0468\frac{F_{b}}{F_{g}}} +} \\{{0.06972\left( {\frac{F_{b}}{F_{g}} - 0.25} \right)^{2}} + {0.01148\left( \frac{F_{b}}{F_{g}} \right)^{6}}}\end{matrix}}} & {{Equation}\mspace{14mu} 11B}\end{matrix}$

The normalized bias voltage V_(DC) _(_) _(n) is related to the biasvoltage V_(DC) as shown in Equation 12. The normalized bias voltageV_(DC) _(_) _(n) is approximately 1.4×10⁻⁸ V_(DC) (meters) for a sealedgap 302 containing vacuum when the ambient pressure is SAP.

$\begin{matrix}{V_{DC\_ n} = {\frac{3}{2}\sqrt{\frac{ɛ_{0}}{P_{0}}}V_{DC}}} & {{Equation}\mspace{14mu} 12}\end{matrix}$

The radiation plate 310 thickness t_(m) is related to the normalizedstatic mechanical force F_(b)/F_(g) and the relative bias levelV_(DC)/V_(C) using the normalized radiation plate 310radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}$

and the normalized bias voltage V_(DC) _(_) _(n) (see Equation 12), asshown in Equation 13.

$\begin{matrix}{t_{m} = {5\; {V_{DC\_ n}\left( \frac{a}{t_{m}} \right)}_{N}^{- 4}\left( \frac{V_{DC}}{V_{C}} \right)^{- 1}\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{3/2}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

The normalized radiation plate 310 radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}$

is related to the radiation plate 310 radius-to-thickness ratio

$\frac{a}{t_{m}}$

as shown in Equation 14. The non-dimensional scaling constant

$\sqrt[4]{\frac{16Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}$

used in Equation 14 is dependent on the elastic properties of theradiation plate 310 (Young's modulus Y₀ and Poisson's ratio σ) and thestatic pressure difference P₀ between the gap 302 and the ambient.

$\begin{matrix}{\left( \frac{a}{t_{m}} \right)_{N} = {\frac{a}{t_{m}}\left( \sqrt[4]{\frac{16Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}} \right)^{- 1}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

The radiation plate 310 thickness t_(m) can also be written in terms ofthe normalized radiation plate 310 thickness t_(m) _(_) _(n) as shown inEquation 15.

$\begin{matrix}{t_{m} = {5{V_{D\; C\; \_ \; n}\left( \frac{V_{D\; C}}{V_{C}} \right)}^{- 1}t_{m\; \_ \; n}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

The normalized radiation plate 310 thickness t_(m) _(_) _(n) is definedin Equation 16 in terms of the normalized radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}$

and the normalized static mechanical force F_(b)/F_(g). The ratio of thecollapse voltage to the collapse voltage in a vacuum V_(C)/V_(r) can besubstituted for using Equation 8. Note that there is an inverserelationship between the size of the normalized radiation plate 310thickness t_(m) _(_) _(n) and the normalized ratio between the gap 302radius and the radiation plate 310 thickness

$\begin{matrix}{{\left( \frac{a}{t_{m}} \right)_{N}.t_{m\; \_ \; n}} = {\left( \frac{a}{t_{m}} \right)_{N}^{- 4}\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{3/2}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

The gap 302 radius a is determined, as shown in Equation 18, using thenormalized gap 302 radius a_(n). The normalized gap 302 radius a_(n) isdefined in Equation 17 in terms of the normalized radiation plate 310radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}.$

The ratio between the collapse voltage and the collapse voltage in avacuum V_(C)/V_(r) can be substituted for using Equation 8. Note thatthere is an inverse relationship between the normalized gap 302 radiusa_(n) and the normalized ratio between the gap 302 radius and theradiation plate 310 thickness

$\begin{matrix}{{\left( \frac{a}{t_{m}} \right)_{N}.a_{n}} = {\left( \frac{a}{t_{m}} \right)_{N}^{- 3}\left\lbrack {\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{3/2}} \right\rbrack}} & {{Equation}\mspace{14mu} 17}\end{matrix}$

As shown in Equation 18, gap 302 radius a is determined by the elasticconstants of a selected radiation plate 310 material, and the operatingpoint. The normalized bias voltage V_(DC) _(_) _(n) is given in Equation12.

$\begin{matrix}{a = {{\left( \sqrt[4]{\frac{16Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}} \right)\left( \frac{a}{t_{m}} \right)_{N}t_{m}} = {\left( {10\sqrt[4]{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right){V_{D\; C\; \_ \; n}\left( \frac{V_{D\; C}}{V_{C}} \right)}^{- 1}a_{n\;}}}} & {{Equation}\mspace{14mu} 18}\end{matrix}$

The equations set forth herein, particularly (but not only) Equations 8,11A, 16, 17 and 19-21, show that the normalized bias voltage V_(DC) _(_)_(n), the normalized gap height t_(ge) _(_) _(n), the normalized gap 302radius a_(n), the normalized radiation plate 310 thickness t_(m) _(_)_(n) and the normalized radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}$

are independent of material properties.

Boundary conditions are discussed below for the radius-to-thicknessratio, gap 302 radius a, radiation plate 310 thickness t_(m), andcorresponding normalized values, with respect to FIGS. 7, 8A and 8B andEquations 19 through 25. Scalability of the radius-to-thickness ratiowhile maintaining optimum sensitivity at a selected operating point isdiscussed below with respect to Equations 26 through 29. Together, theseFigures and Equations demonstrate a certain degree of flexibility inselection of particular MCM 300 dimensions for use at a selectedoperating point while maintaining optimum sensitivity (subject to therelationships described herein).

FIG. 7 shows a lin-log semi-log graph 700 of the relationship between amaximum normalized radiation plate 310 radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N\; \_ \; {ma}\; x}$

that enables an MCM 300 to meet the elastic linearity constraint(explained below), and normalized static mechanical force F_(b)/F_(g),for example values of the relative bias voltage level V_(DC)/V_(C). Theelastic linearity constraint can be explained using Hooke's law. Hooke'slaw defines the behavior of linearly elastic structures under stress.Hooke's law states that the displacement in a spring (or other linearlyelastic structure) is proportional to a force which stretches orcompresses it. If the force is doubled, the displacement of the springwill be doubled. However, once a real spring is sufficiently displaced(stretched), doubling the force will not double the displacement,deviating from Hooke's law. This is due to elastic non-linearity. Beyondan upper bound for applied force and for displacement, Hooke's law nolonger holds and the relationship between applied force and springdisplacement is no longer linear.

Hooke's law applies to clamped membranes as long as linearly elasticoperation holds. Studies on applied mechanics classify the linearlyelastic range, that is, the displacement range in which Hooke's law isapplicable to a clamped circular radiation plate, as corresponding tothe center deflection of the radiation plate being less than 20% of theplate thickness, that is, X_(P)/t_(m)<0.2. The sensitivity of an MCM 300will decrease when elastic linearity fails, accordingly, whenX_(P)/t_(m)≥0.2. This limit for linearly elastic behavior of an MCM 300is referred to herein as the “elastic linearity constraint.”

The elastic linearity constraint can be used to determine a maximumvalue for the radiation plate 310 radius-to-thickness ratio a/t_(m) atwhich an MCM 300 at a particular operating point will exhibit linearlyelastic behavior. This maximum radius-to-thickness ratio a/t_(m)corresponds to minima for the gap 302 radius a and the radiation plate310 thickness t_(m). Accordingly, as shown in Equations 12 and 15-18herein, there is an inverse relationship between (1) the size of the gap302 radius a and radiation plate 310 thickness t_(m) (the radiationplate 310 dimensions), and (2) the radiation plate 310radius-to-thickness ratio a/t_(m).

Elastic linearity of CMUT cells is described in A. Unlugedik, A. S.Tasdelen, A. Atalar, and H. Köymen, “Designing Transmitting CMUT Cellsfor Airborne Applications,” IEEE Transactions on Ultrasonics,Ferroelectrics, and Frequency Control, Vol. 61, pp. 1899-1910, 2014,which is incorporated herein by reference.

The maximum radiation plate 310 radius-to-thickness ratio a/t_(m) isfound using a maximum normalized radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N\; \_ \; {ma}\; x},$

which is related to the normalized static displacement of the center ofthe radiation plate 310 X_(P)/t_(ge) as shown in Equation 19. InEquation 19, the normalized static displacement of the center of theradiation plate 310 X_(P)/t_(ge) is expressed as

${X_{PN}\left( {\frac{V_{D\; C}}{V_{C}},\frac{F_{b}}{F_{g}}} \right)},$

that is, as a function of the relative bias V_(DC)/V_(C) and thenormalized static mechanical force F_(b)/F_(g).

$\begin{matrix}{{\left( \frac{a}{t_{m}} \right)_{N} < \left( \frac{a}{t_{m}} \right)_{N\; \_ \; m\; {ax}}} = \sqrt[4]{\frac{F_{b}}{F_{g}}/{X_{PN}\left( {\frac{V_{D\; C}}{V_{C}},\frac{F_{b}}{F_{g}}} \right)}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

The normalized static displacement of the center of the radiation plate310

$X_{PN}\left( {\frac{V_{DC}}{V_{C}},\frac{F_{b}}{F_{g}}} \right)$

is obtained in Equation 20 by solving Equation 7, and substituting forthe collapse voltage in a vacuum V_(r) using Equation 8.

$\begin{matrix}{{{\begin{matrix}{\frac{V_{DC}^{2}}{V_{C}^{2}}\left( {0.9961 - {1.0468\frac{F_{b}}{F_{g}}} + {0.06972\left( {\frac{F_{b}}{F_{g}} - 0.25} \right)^{2}} +} \right.}\end{matrix}\begin{matrix}{{{\left. {0.01148\left( \frac{F_{b}}{F_{g}} \right)^{6}} \right)^{2}2{g^{\prime}\left( \frac{X_{P}}{t_{ge}} \right)}} -}\mspace{11mu}}\end{matrix}3\left( {\frac{X_{P}}{t_{ge}} - \frac{F_{b}}{F_{g}}} \right)} \approx 0}{{{for}\mspace{14mu} \frac{X_{P}}{t_{ge}}} \geq \frac{F_{b}}{F_{g}}}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

As shown in FIG. 7,

${\left( \frac{a}{t_{m}} \right)_{N_{-}\max} \leq 1};$

that is, the maximum value of the maximum normalized radius-to-thicknessratio

$\left( \frac{a}{t_{m}} \right)_{N_{-}\max},$

which is unity (one), is reached at F_(b)/F_(g)=1 (see description ofFIG. 4 and Equation 6 regarding normalized static mechanical forceF_(b)/F_(g)). The maximum normalized radius-to-thickness ratio(a/t_(m))_(N) _(_) _(max) that enables an MCM 300 to meet the elasticlinearity constraint decreases as normalized static mechanical forceF_(b)/F_(g) decreases or as relative bias level V_(DC)/V_(C) decreases.

The scaling constant term

$\sqrt[4]{\frac{16Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}$

relating radius a to normalized radius a_(n) (see Equations 14 and 18)is 35.6 for a silicon radiation plate 310, if Young's modulus Y₀ is149×10⁹ Pa, Poisson's ratio σ is 0.17, and the static pressuredifference P₀ between the gap 302 and the ambient equals SAP (101.325kPa). For a silicon radiation plate 310, the radius-to-thickness ratioa/t_(m) will therefore, to maintain linearly elastic operation, be keptless than 35.6 at large normalized static mechanical force F_(b)/F_(g)if the static pressure differential P₀ is equal to SAP. This upper limitfor radius-to-thickness ratio a/t_(m) for elastic linear operationdecreases as the normalized static mechanical force F_(b)/F_(g)decreases. The minimum radius-to-thickness ratio a/t_(m) is about 8 forF_(b)/F_(g)=0.001. Because there is an inverse relationship between theradiation plate dimensions (a and t_(m)) and the radius-to-thicknessratio a/t_(m), the elastic linearity constraint suggests that the lowerthe normalized static mechanical force F_(b)/F_(g), the larger the gap302 radius a should be.

Accordingly, a maximum normalized radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N_{-}\max}$

implies minimum values for the gap 302 radius a_(min) and the radiationplate 310 thickness t_(m) _(_) _(min) that enable an MCM 300 to operatein the linearly elastic regime at a selected operating point(F_(b)/F_(g), V_(DC), V_(DC)/V_(C)). The minimum gap 302 radius a_(min)corresponds to the narrowest gap 302 that enables linearly elasticoperation at a selected operating point. The minimum radiation plate 310thickness t_(m) _(_) _(min) corresponds to the thinnest radiation plate310 that enables linearly elastic operation at a selected operatingpoint.

Equation 21 shows the relationship between minimum radius a_(min) andnormalized minimum radius a_(n) _(_) _(min), which is found using therelationship between gap 302 radius a and normalized gap 302 radiusa_(n) as described by Equation 18.

$\begin{matrix}{a_{\min} = {\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right){V_{{DC}_{-}n}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}a_{n_{-}\min}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

FIG. 8A shows a log-lin semi-log graph 800 of the relationship betweenminimum normalized gap 302 radius a_(n) _(_) _(min) that enables an MCM300 to meet the elastic linearity constraint, and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C). Equation 22 defines normalized minimum gap302 radius a_(n) _(_) _(min) in terms of maximum normalized radiationplate 310 radius-to-thickness ratio (a/t_(m))_(N) _(_) _(max), using therelationship between normalized gap 302 radius a_(n) and normalizedradiation plate 310 radius-to-thickness ratio (a/t_(m))_(N) as describedby Equation 17. Note that the ratio between the clamp voltage and thecollapse voltage in a vacuum V_(C)/V_(r) depends only on the normalizedstatic mechanical force F_(b)/F_(g), as shown in Equation 8.

$\begin{matrix}{a_{n_{-}\min} = {\left( \frac{a}{t_{m}} \right)_{N_{-}\max}^{- 3}\left\lbrack {\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{3\text{/}2}} \right\rbrack}} & {{Equation}\mspace{14mu} 22}\end{matrix}$

Equation 23 shows the relationship between minimum radiation plate 310thickness t_(m) _(_) _(min) and normalized minimum radiation plate 310thickness t_(m) _(_) _(n) _(_) _(min), which is found using therelationship between radiation plate 310 thickness t_(m) and normalizedradiation plate 310 thickness t_(m) _(_) _(n) as described by Equation15.

$\begin{matrix}{t_{m_{-}\min} = {5{V_{{DC}_{-}n}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}t_{m_{-}n_{-}\min}}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

FIG. 8B shows a log-lin semi-log graph 802 of the relationship betweennormalized minimum radiation plate 310 thickness t_(m) _(_) _(n) _(_)_(min) that enables an MCM 300 to meet the elastic linearity constraint,and normalized static mechanical force F_(b)/F_(g), for example valuesof the relative bias voltage level V_(DC)/V_(C). Equation 24 definesnormalized minimum radiation plate 310 thickness t_(m) _(_) _(n) _(_)_(min) in terms of maximum normalized radius-to-thickness ratio(a/t_(m))_(N) _(_) _(max), using the relationship between normalizedminimum radiation plate 310 thickness t_(m) _(_) _(min) and normalizedradius-to-thickness ratio (a/t_(m))_(N) as described by Equation 16.Note that the ratio between the clamp voltage and the collapse voltagein a vacuum V_(C)/V_(r) depends only on the normalized static mechanicalforce F_(b)/F_(g), as shown in Equation 8.

$\begin{matrix}{t_{m_{-}n_{-}\min} = {{a_{n_{-}\min}\left( \frac{a}{t_{m}} \right)}_{N_{-}\max}^{- 1} = {\left( \frac{a}{t_{m}} \right)_{N_{-}\max}^{- 4}\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{3\text{/}2}}}} & {{Equation}\mspace{14mu} 24}\end{matrix}$

The scaling constant term in Equation 21 is determined for silicon inEquation 25, taking Young's modulus Y₀ to be 149×10⁹ Pa, Poisson's ratioσ to be 0.17, and the static pressure difference P₀ between the gap 302and the ambient to equal SAP (101.325 kPa).

$\begin{matrix}{{10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} = 177.95} & {{Equation}\mspace{14mu} 25}\end{matrix}$

This normalization parameter for the minimum gap 302 radius a_(min) isnon-dimensional and contains only the elastic constants of the radiationplate 310 material and the differential static pressure P₀. Thenormalized minimum gap 302 radius a_(n) _(_) _(min) and radiation plate310 thickness t_(m) _(_) _(n) _(_) _(min) are independent of materialand ambient physical properties and the bias voltage V_(DC). Thenormalized minimum gap 302 radius a_(n) _(_) _(min) and radiation plate310 thickness t_(m) _(_) _(n) _(_) _(min) are instead determined bynormalized static mechanical force F_(b)/F_(g) and relative bias voltageV_(DC)/V_(C), as shown in Equations 8, 19, 20, 22 and 24.

Using Equations 26-29, a normalized radiation plate 310radius-to-thickness ratio (a/t_(m))_(N) can be chosen (within thelimitations described by the equations) that is less than the maximumradius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N_{-}\max},$

that is, less than the value of the normalized radius-to-thickness ratio(a/t_(m))_(N) at the elastic linearity limit. The smaller the normalizedradius-to-thickness ratio (a/t_(m))_(N) of an MCM 300 operated at aselected operating point (F_(b)/F_(g), V_(DC), V_(DC)/V_(C)), the largerthe gap 302 radius a, the thicker the radiation plate 310 (largert_(m)), and the more robust the linearly elastic operation (less proneto variations in operation removing the MCM 300 from the linearlyelastic regime) of the MCM 300 operated at the selected operating point;without changing the OCRV sensitivity corresponding to that operatingpoint. Further, increased normalized radius-to-thickness ratio(a/t_(m))_(N) (within the limitations described in the equations)results in increased input capacitance C_(in) of the MCM 300, which isadvantageous for pre-amplification electronics. Also, the larger theclamped capacitance C₀, the smaller the relative effect of parasiticcapacitance on MCM 300 performance. Accordingly, a choice of normalizedradius-to-thickness ratio (a/t_(m))_(N) can be made while retaining thesame optimal OCRV sensitivity at the selected operating point.

A scalar K is defined in Equation 26, relating the normalizedradius-to-thickness ratio (a/t_(m))_(N) to the maximum normalizedradius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N_{-}\max}.$

K, as expressed in Equation 27, is defined to satisfy the elasticlinearity constraint. Accordingly, K is larger than unity, that is, K>1.

$\begin{matrix}{\left( \frac{a}{t_{m}} \right)_{N} = {\frac{1}{K}\left( \frac{a}{t_{m}} \right)_{N_{-}\max}}} & {{Equation}\mspace{14mu} 26} \\{\left( \frac{a}{t_{m}} \right)_{N} < \left( \frac{a}{t_{m}} \right)_{N_{-}\max}} & {{Equation}\mspace{14mu} 27}\end{matrix}$

The normalized radius a_(n) can be expressed in terms of the minimumnormalized gap 302 radius a_(n) _(_) _(min) and the scalar K as shown inEquation 28. The normalized thickness t_(m) _(_) _(n) of the radiationplate 310 can be expressed in terms of the minimum normalized thicknessof the radiation plate 310 t_(m) _(_) _(n) _(_) _(min) and the scalar Kas shown in Equation 29. A larger K means a radiation plate 310 that isthicker relative to the gap 302 radius a. There is an upper limit for K,approximately K<5, above which the radiation plate 310 becomes too thickfor the model to be valid. Further, in some embodiments comprising anMCM 300 fabricated from typical materials and intended for use in an airenvironment, K<2.5 is preferable. Microphones with K over 2.5 will havegap 302 radius a much larger than radiation plate 310 thickness t_(m).This can make the MCM 300 difficult and/or expensive to manufacture, andpotentially fragile in operation.

a _(n)=(K ³)a _(n) _(_) _(min)   Equation 28

t _(m) _(_) _(n)=(K ₄)t _(m) _(_) _(n) _(_) _(min)   Equation 29

For a particular selected operating point triplet

$\left( {\frac{F_{b}}{F_{g}},V_{DC},\frac{V_{DC}}{V_{C}}} \right),$

changes in K (within boundaries as described) will not affect the MCM300 sensitivity or the effective gap 302 height t_(ge).

Open Circuit Receive Voltage (OCRV) sensitivity of an MCM 300 isobtained, in volts (V) per pascal (Pa), as shown in Equation 30.(Particular units are used herein by way of example only; other unitscan be used.) The OCRV sensitivity is represented by S_(VO).S_(VO)=V_(OC)/p. V_(OC) is the voltage across the electrical terminals(not shown) of the MCM 300 when the terminals are in open circuit, and prepresents incident pressure, meaning that V_(OC)/p describes thestrength (V_(OC)) of the voltage induced between the terminals of amicrophone circuit by a pressure wave of magnitude p incident on theradiation plate 310. Equation 30 assumes that the MCM 300 is mounted ona rigid baffle and operated off-resonance, and ignores radiationimpedance (losses from radiation impedance are discussed in theBackground, above).

$\begin{matrix}{S_{VO} = {\frac{V_{oc}}{p} = {{- \left\lbrack {\frac{3}{8}\sqrt{\frac{2}{5}}\sqrt{\frac{1 - \sigma^{2}}{ɛ_{0}Y_{0}}}\left( \frac{a^{2}}{t_{m}} \right)\sqrt{\frac{t_{ge}}{t_{m}}}} \right\rbrack}{h\left( {\frac{X_{P}}{t_{ge}},\frac{F_{b}}{F_{g}},\frac{C_{p}}{C_{0}}} \right)}\frac{V}{Pa}}}} & {{Equation}\mspace{14mu} 30}\end{matrix}$

Equation 30 can be rewritten so that OCRV sensitivity is expressed interms of the operating point parameters (F_(b)/F_(g), V_(DC),V_(DC)/V_(C)). This is done using expressions for effective gap 302height t_(ge), radiation plate 310 thickness t_(m), and gap 302 radiusa, in Equations 10, 15 and 18, respectively. Expressions for inputcapacitance C_(in) and clamped capacitance C₀ in terms of the operatingpoint are provided in Equations 37 and 38, respectively.

$\begin{matrix}{S_{VO} = {\frac{V_{oc}}{p} = {{- \left\lbrack {\frac{9}{2}\sqrt{\frac{1}{5}}\left( \frac{V_{DC}}{P_{0}} \right)} \right\rbrack}{h_{oc}\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}},\frac{C_{p}}{C_{0}}} \right)}\frac{V}{Pa}}}} & {{Equation}\mspace{14mu} 31}\end{matrix}$

The dimensionless normalized OCRV sensitivity h_(oc) is given as shownin Equation 32. The dimensionless normalized OCRV sensitivity h_(oc) isa function of the parasitic capacitance C_(p) and the operating pointparameters voltage bias level V_(DC)/V_(C) and normalized staticmechanical force F_(b)/F_(g). The functions g(u), g′(u), and g″(u) areshown and described with respect to Equations 2-4 (above). Preferably,the parasitic capacitance C_(p) is relatively small compared to theinput capacitance C₀, for example, small enough that the effects of theparasitic capacitance can be ignored and/or do not prevent meetingdesign performance specifications. The ratio of the collapse voltage tothe collapse voltage in a vacuum V_(C)/V_(r) can be substituted forusing Equation 8. The dimensionless normalized OCRV sensitivity h_(oc)is evaluated at the static equilibrium shown in Equation 20, but doesnot explicitly depend on the dimensions of the MCM 300 (such as gap 302radius a) or material properties (such as Poisson's ratio). Thenormalized static displacement of the center of the radiation plate 310X_(PN) equals the ratio of the static displacement X_(P) to theeffective gap height t_(ge) at the operating point, as shown in Equation33. Also, as shown in Equation 33, the normalized static displacementX_(PN) depends only on the relative bias level V_(DC)/V_(C) and thenormalized static mechanical force F_(b)/F_(g).

$\begin{matrix}{{h_{oc}\left( {\frac{V_{DC}}{V_{C}},\frac{F_{b}}{F_{g}},\frac{C_{p}}{C_{0}}} \right)} = \frac{\frac{F_{b}}{F_{g}}{g^{\prime}\left( X_{PN} \right)}}{\begin{matrix}{\left( {\frac{V_{DC}}{V_{C}}\frac{V_{C}}{V_{r}}{g^{\prime}\left( X_{PN} \right)}} \right)^{2} +} \\{\left\lbrack {\frac{C_{p}}{C_{0}} + {g\left( X_{PN} \right)}} \right\rbrack \left\lbrack {\frac{3}{4} - {\frac{1}{2}\left( {\frac{V_{DC}}{V_{C}}\frac{V_{C}}{V_{r}}} \right)^{2}{g^{''}\left( X_{PN} \right)}}} \right\rbrack}\end{matrix}}} & {{Equation}\mspace{14mu} 32} \\{\mspace{76mu} {X_{PN} = {\frac{X_{P}}{t_{ge}}\mspace{14mu} {at}\mspace{14mu} {the}\mspace{14mu} {operating}\mspace{14mu} {point}\mspace{14mu} \left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right)}}} & {{Equation}\mspace{14mu} 33}\end{matrix}$

The OCRV sensitivity is a linear function of the ratio of the biasvoltage to the static pressure difference between the gap 302 and theambient V_(DC)/P₀. The sensitivity coefficient given in Equation 31 canbe restated, using Equation 12 and holding the static pressuredifferential P₀ to be SAP, as shown in Equation 34.

$\begin{matrix}{{\frac{9}{2}\sqrt{\frac{1}{5}}\left( \frac{V_{DC}}{P_{0}} \right)} \approx \left\{ \begin{matrix}{{2 \times 10^{- 5}V_{DC}}\mspace{220mu}} & \frac{V}{Pa} \\{{\frac{3}{\sqrt{5ɛ_{0}P_{0}}}V_{{DC}_{—}n}} = {1416.5V_{{DC}_{—}n}}} & \frac{V}{Pa}\end{matrix} \right.} & {{Equation}\mspace{14mu} 34}\end{matrix}$

As shown in Equation 34, the sensitivity coefficient can be described as2×10⁻⁵ V/Pa per volt bias when the static pressure differential P₀ isSAP. Equations 6 and 7 show that the OCRV sensitivity is indirectlyrelated to (though, as shown herein, not dependent on) the materialproperties of the radiation plate 310 through the normalized staticmechanical force F_(b)/F_(g) and V_(DC). As a result, it can be seenthat sensitivity increases (improves) as F_(b)/F_(g) and/or V_(DC)increases, and sensitivity decreases (worsens) as F_(b)/F_(g) and/orV_(DC) decreases.

The sensitivity of an MCM with a gap in the shape of a regular convexpolygon (such as a square, pentagon, hexagon or octagon) with area equalto a circle of a particular radius R can be modelled similarly to an MCM300 with a gap 302 of radius R, with additional parallel inputcapacitance—similar to parasitic capacitance—as part of the model.Because of the additional parasitic capacitance, such non-circulargeometries will generally have lower OCRV sensitivity than an MCM 300with a circular gap 302.

An MCM with an elliptical gap can also be modelled similarly to an MCMwith a circular gap; as can a convex polygon with a same aspect ratioand equal gap area as an ellipse (such as a rectangle); with additionalparallel input capacitance as part of the model. Because of theadditional parasitic capacitance, such non-circular geometries willgenerally have lower OCRV sensitivity than an MCM 300 with a circulargap 302.

Irregular convex polygons and concave polygons can also be modelled byan equivalent circle with an area smaller than the area of the polygon,and with a parallel capacitance (in this case, resulting in asignificantly higher ratio between parasitic capacitance C_(p) andclamped capacitance C₀). Because of the additional parasiticcapacitance, such non-circular geometries will generally have lower OCRVsensitivity than an MCM 300 with a circular gap 302, or a gap in theshape of a regular polygon.

In other words, an equivalent circular gap can be defined for apolygonal gap geometry, using an additional parallel capacitance toadapt the circular gap model described herein to the different geometry.

FIG. 9 shows a graph 900 of the relationship between normalized OpenCircuit Received Voltage Sensitivity (OCRV) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C), where parasitic capacitance C_(p) divided byclamped capacitance C₀ equals zero, that is, C_(p)/C₀=0. Thisrelationship is provided in Equation 35, which is obtained usingEquations 12 and 31 (as described with respect to Equation 34). For anMCM 300 with a gap 302 containing a vacuum, when the ambient pressure isSAP, the static pressure difference between the gap 302 and the ambientis P₀=101.325 kPa.

$\begin{matrix}\begin{matrix}{S_{{VO}_{—}n} = {S_{VO} - {20\mspace{14mu} \log \mspace{14mu} V_{{DC}_{—}n}}}} \\{= \mspace{14mu} \begin{matrix}{\mspace{14mu} \left. {20\mspace{14mu} \log} \middle| {\frac{3}{\sqrt{5ɛ_{0}P_{0}}}{h_{oc}\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}},0} \right)}} \right|} \\{{dB}\mspace{14mu} {re}\mspace{14mu} \frac{V}{{Pa} \times m}}\end{matrix}}\end{matrix} & {{Equation}\mspace{14mu} 35}\end{matrix}$

As shown in FIG. 9, normalized OCRV sensitivity varies less than 5 dBfor relative bias voltage levels V_(DC)/V_(C) between 0.4 and 0.9, andfor possible levels of normalized static mechanical force F_(b)/F_(g)(as described above with respect to FIG. 4 and Equation 6). Also, asshown in FIG. 9, the higher the normalized static mechanical forceF_(b)/F_(g), the higher the normalized OCRV sensitivity.

At the elastic linearity threshold, that is, when a=a_(min) andt_(m)=t_(m) _(_) _(min), the sensitivity is about 1 dB less than theOCRV sensitivity given in Equation 35. This is related to the elasticlinearity constraint being an approximation (there is generally not asudden transition in microphone performance characteristics at theboundary of the elastic linearity constraint as described herein). Whenthe radius-to-thickness ratio a/t_(m) is lower than the maximum, theradiation plate 310 is relatively thicker and the MCM 300 maintains theOCRV sensitivity corresponding to the operating point, as described inEquation 35.

Advantageously, increasing clamp capacitance and input capacitancereduces the effect of parasitic capacitance on OCRV sensitivity, andenables better performance in front-end electronics designs.Accordingly, if input capacitance Cin is large compared to parasiticcapacitance, then the amount by which the parasitic capacitance reducesthe OCRV sensitivity will be diminished (or eliminated). Also, ifclamped capacitance is increased, microphone impedance will be lowered;in some embodiments, this can enable simpler pre-amplifier design,higher pre-amplifier gain, and lower pre-amplifier noise contribution.The deflected clamped capacitance C_(0d) (clamped capacitance when theradiation plate 310 is deflected by the static deflection X_(P)) at theoperating point (F_(b)/F_(g), V_(DC), V_(DC)/V_(C)) is related to theclamped capacitance C₀ as shown in Equation 36. The input capacitanceC_(in) at the operating point is given in Equation 37 (see Equations2-4).

$\begin{matrix}{C_{0d} = {C_{0}{g\left( X_{PN} \right)}}} & {{Equation}\mspace{14mu} 36} \\{C_{in} = {C_{0}\mspace{14mu} \left\{ {{g\left( X_{PN} \right)} + {\frac{4}{3}\frac{{\frac{V_{DC}^{2}V_{C}^{2}}{V_{C}^{2}V_{r}^{2}}\left\lbrack {g^{\prime}\left( X_{PN} \right)} \right\rbrack}^{2}}{1 - {\frac{{{}_{}^{\;}{}_{}^{}}V_{C}^{2}}{3V_{C}^{2}V_{r}^{2}}{g^{''}\left( X_{PN} \right)}}}}} \right\}}} & {{Equation}\mspace{14mu} 37}\end{matrix}$

The normalized static displacement X_(PN) is determined using Equation33. The clamped capacitance C₀ for an MCM 300 with a radiation plate 310radius-to-thickness ratio

$\frac{a}{t_{m}}$

and operating in the linearly elastic regime is expressed in terms ofthe operating point parameters as shown in Equation 38. The clampedcapacitance C₀ equals the area of the MCM 300 cell divided by theeffective gap height t_(ge), C₀=πa²/t_(ge). Equation 38 is producedusing this relationship, and using Equations 9, 17 and 18. The ratio ofthe collapse voltage to the collapse voltage in a vacuum V_(C)/V_(r) canbe substituted for using Equation 8. The physical constant-dependentmultiplier in Equation 38 has units of farads.

$\begin{matrix}{C_{0} = {{\pi ɛ}_{0}\mspace{14mu} \left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)^{2}{V_{{DC}_{—}n}\left( \frac{a}{t_{m}} \right)}_{N}^{- 6}\left( \frac{V_{DC}}{V_{C}} \right)^{- 1}\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{5\text{/}2}}} & {{Equation}\mspace{14mu} 38}\end{matrix}$

As shown in Equation 38, C₀ is inversely proportional to the sixth powerof the radius-to-thickness ratio

$\frac{a}{t_{m}}.$

When the normalized radius-to-thickness ratio (a/t_(m))_(N) is chosen tobe

${0.794 \times \left( \frac{a}{t_{m}} \right)_{N_{—}\max}},$

corresponding to K=1.26, the gap radius 302 a is doubled (see Equation28) and the input capacitance C_(in) is increased by a factor of four.As described above, because this does not change the operating point andobeys the elastic linearity constraint, it also does not change the OCRVsensitivity.

Equation 39 shows the physical constant-dependent multiplier for asilicon radiation plate 310, where differential static pressure P₀equals SAP.

$\begin{matrix}{{{{\pi ɛ}_{0}\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)}^{2}V_{{DC}_{—}n}} = {12.33 \times 10^{- 15}V_{DC}\mspace{14mu} F}} & {{Equation}\mspace{14mu} 39}\end{matrix}$

FIG. 10 shows a log-lin semi-log graph 1000 of the relationship betweennormalized input capacitance C_(in) _(_) _(n) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C), where the normalized radius-to-thicknessratio equals the maximum normalized radius-to-thickness ratio

$\left( \frac{a}{t_{m}} \right)_{N}$

which enables linearly elastic operation. The normalized inputcapacitance C_(in) _(_) _(n) is shown in Equation 40 in terms of theoperating point. Equation 8 can be used to substitute for the ratiobetween the collapse voltage and the collapse voltage in a vacuumV_(C)/V_(r).

$\begin{matrix}{C_{{in}_{—}n} = {\left\{ {{g\left( X_{PN} \right)} + {\frac{4}{3}\frac{\frac{V_{DC}^{2}}{V_{C}^{2}}{\frac{V_{C}^{2}}{V_{r}^{2}}\left\lbrack {g^{\prime}\left( X_{PN} \right)} \right\rbrack}^{2}}{1 - {\frac{2}{3}\frac{V_{DC}^{2}}{V_{C}^{2}}\frac{V_{C}^{2}}{V_{r}^{2}}{g^{''}\left( X_{PN} \right)}}}}} \right\} \left( \frac{a}{t_{m}} \right)_{N}^{- 6}\left( \frac{V_{DC}}{V_{C}} \right)^{- 1}\left( \frac{V_{C}}{V_{r}} \right)^{- 1}\left( \frac{F_{b}}{F_{g}} \right)^{5\text{/}2}}} & {{Equation}\mspace{14mu} 40}\end{matrix}$

Using Norton source transformation and Equation 31, the SCRC sensitivitycan be obtained from the OCRV sensitivity as shown in Equation 41. TheSCRC sensitivity is represented by S_(IS). S_(IS)=I_(SC)/p. I_(SC) isshort circuit current, and p represents incident pressure, meaning thatI_(SC)/p describes the strength (S_(IS)) of the current induced betweenthe shorted terminals of an MCM 300 by a pressure wave of magnitude pincident on the radiation plate 310. The SCRC sensitivity is related tothe OCRV sensitivity according to I_(SC)=−jωC_(in)V_(OC). Here, corepresents the radial frequency of the sound signal at which thesensitivity is evaluated. The (−jω) portion of the expression means thatthe SCRC sensitivity increases as the frequency increases.

$\begin{matrix}{S_{IS} = {\frac{I_{sc}}{p} = {{- j}\; \omega \; {C_{in}\left\lbrack {\frac{9}{2}\sqrt{\frac{1}{5}}\left( \frac{V_{DC}}{P_{0}} \right)} \right\rbrack}{h_{oc}\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}},\frac{C_{p}}{C_{0}}} \right)}\mspace{14mu} \frac{A}{Pa}}}} & {{Equation}\mspace{14mu} 41}\end{matrix}$

Equation 41 can be rewritten to obtain Equation 42, using Equations 37,38 and 40. Equation 43 shows the expression for SCRC sensitivity ofEquation 42, in units of dB re

$\frac{A}{Pa},$

corresponding to decibels relative to amps per pascal.

$\begin{matrix}{\frac{I_{sc}}{p} = {{- j}\; {{\omega\pi ɛ}_{0}\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)}^{2}V_{{DC}_{—}n}C_{{in}_{—}n}\left\{ {\left\lbrack {\frac{9}{2}\sqrt{\frac{1}{5}}\left( \frac{V_{DC}}{P_{0}} \right)} \right\rbrack {h_{oc}\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}},\frac{C_{p}}{C_{0}}} \right)}} \right\} \mspace{14mu} \frac{A}{Pa}}} & {{Equation}\mspace{14mu} 42} \\\begin{matrix}{S_{IS} = {20\mspace{14mu} \log \frac{I_{sc}}{p}\mspace{14mu} {dB}\mspace{14mu} {re}\mspace{14mu} \frac{A}{Pa}}} \\{= {\left\{ {S_{VO} - {20\mspace{14mu} \log \mspace{14mu} V_{{DC}_{—}n}}} \right\} + {20\mspace{14mu} \log \mspace{14mu} C_{{in}_{—}n}} + {40\mspace{14mu} \log \mspace{14mu} V_{{DC}_{—}n}} + {20\mspace{14mu} \log \mspace{14mu} \left( {{\omega\pi ɛ}_{0}\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)}^{2} \right)}}}\end{matrix} & {{Equation}\mspace{14mu} 43}\end{matrix}$

Equation 44 expresses the second term of Equation 43 using thecorresponding operating point parameter, bias voltage V_(DC). Equation45 provides the value of the fifth (last) term of Equation 43 at 1 kHzoperating frequency for a crystalline silicon radiation plate 310 atSAP. The unit S is Siemens.

$\begin{matrix}{\mspace{76mu} {{20\mspace{14mu} \log \mspace{14mu} V_{{DC}_{—}n}} = {{- 157.1} + {20\mspace{14mu} \log \mspace{14mu} V_{DC}\mspace{14mu} {dB}\mspace{14mu} {re}\mspace{14mu} (m)}}}} & {{Equation}\mspace{14mu} 44} \\{{20\mspace{14mu} \log \mspace{14mu} \left\{ {{\omega\pi ɛ}_{0}\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)}^{2} \right\}} = {{- 45.1}\mspace{14mu} {dB}\mspace{14mu} {re}\mspace{14mu} \frac{S}{m}\mspace{14mu} {at}\mspace{14mu} 1\mspace{14mu} {kHz}}} & {{Equation}\mspace{14mu} 45}\end{matrix}$

Equation 46 provides a simplified version of Equation 43, in terms ofbias voltage V_(DC) and normalized input capacitance C_(in) _(_) _(n).

$\begin{matrix}{S_{IS} = {\left\{ {S_{VO} - {20\mspace{14mu} \log \mspace{14mu} V_{DC}}} \right\} + {20\mspace{14mu} \log \mspace{14mu} C_{{in}_{—}n}} + {40\mspace{14mu} \log \mspace{14mu} V_{DC}} - {359.3\mspace{14mu} {dB}\mspace{14mu} \frac{A}{Pa}\mspace{14mu} {at}\mspace{14mu} 1\mspace{14mu} {kHz}}}} & {{Equation}\mspace{14mu} 46}\end{matrix}$

FIG. 11 shows a graph 1100 of the relationship between normalized ShortCircuit Received Current Sensitivity (SCRC_(n)) and normalized staticmechanical force F_(b)/F_(g), for example values of the relative biasvoltage level V_(DC)/V_(C). The normalized SCRC sensitivity isdetermined as shown in Equation 47.

SCRC_(n) sensitivity=S _(IS)−40 log V _(DC) _(_) _(n)   Equation 47

FIG. 12 shows a graph 1200 of the relationship between normalized ShortCircuit Received Current Sensitivity (SCRC_(n)) per square meter andnormalized static mechanical force F_(b)/F_(g), for example values ofthe relative bias voltage level V_(DC)/V_(C). OCRV sensitivity isindependent of the area of the MCM 300 (the area of the circular gap302), whereas SCRC sensitivity depends on the area of the MCM 300. SCRCsensitivity per square meter S_(IS)/m² is obtained by normalizing SCRCto the area of the cell (SCRC is divided by the area of an MCM 300 cellπa²). Equation 48 is obtained using Equations 17, 37, 38, 40 and 41.

$\begin{matrix}\begin{matrix}{{S_{IS}\text{/}m^{2}} = {{20\mspace{14mu} \log \mspace{14mu} \frac{I_{sc}}{\pi \; a^{2}p}} = \left. {20\mspace{14mu} \log}\mspace{14mu} \middle| {{- j}\; \omega {\frac{C_{in}}{\pi \; a^{2}}\left\lbrack {\frac{9}{2}\sqrt{\frac{1}{5}}\left( \frac{V_{DC}}{P_{0}} \right)} \right\rbrack}{h_{oc}\left( {\frac{V_{DC}}{V_{C}},\frac{F_{b}}{F_{g}},\frac{C_{p}}{C_{0}}} \right)}} \middle| \mspace{14mu} {{dB}\mspace{14mu} {re}\mspace{14mu} \frac{A}{{Pa} \times m^{2}}} \right.}} \\{= \left. {20\mspace{14mu} \log}\mspace{14mu} \middle| {{- j}\; {\omega \left( {3\sqrt{\frac{ɛ_{0}}{5P_{0}}}} \right)}\left\{ {\frac{C_{{in}_{—}n}}{a_{n}^{2}}{h_{oc}\left( {\frac{V_{DC}}{V_{C}},\frac{F_{b}}{F_{g}},\frac{C_{p}}{C_{0}}} \right)}} \right\}} \middle| \mspace{14mu} {{dB}\mspace{14mu} {re}\mspace{14mu} \frac{A}{{Pa} \times m^{2}}} \right.}\end{matrix} & {{Equation}\mspace{14mu} 48}\end{matrix}$

The constant term in Equation 48,

${3\sqrt{\frac{ɛ_{0}}{5P_{0}}}},$

can be evaluated as shown in Equation 49, taking the static pressuredifferential P₀ to be SAP.

$\begin{matrix}{{3\sqrt{\frac{ɛ_{0}}{5P_{0}}}} = {{1.254 \times 10^{- 8}\mspace{14mu} \frac{A \times \sec}{{Pa} \times m^{2}}} = {{- 158}\mspace{14mu} {dB}\mspace{14mu} {re}\mspace{14mu} \frac{A \times \sec}{{Pa} \times m^{2}}}}} & {{Equation}\mspace{14mu} 49}\end{matrix}$

SCRC sensitivity per unit area (S_(IS)/m²) is independent of materialproperties and the bias voltage V_(DC), and provides better guidance forthe choice of operational parameters than unmodified SCRC sensitivityS_(IS). This is because, generally, the larger the MCM 300 cell area,the better the sensitivity of the MCM 300 cell.

Sensitivity can also be increased by using multiple MCM 300 cells whichare electrically connected in parallel.

A wide variety of combinations of less than all operating pointparameters can be specified at the beginning of MCM 300 design so thatthe specified values are sufficient to determine the correspondingremaining MCM 300 characteristics. That is, combinations can bespecified of a (small) subset of MCM 300 dimensions, MCM 300 OCRV andSCRC sensitivities, and/or other MCM 300 characteristics, and theremaining MCM characteristics can be determined from the selectedvalues. This is enabled by the relationships between operating pointparameters and MCM 300 properties as described above; as well as by theuse of normalized dimensions, which are independent of properties ofmaterials to be used in MCM manufacture; and by the scaling propertiesdescribed with respect to Equations 26-29, which can be used to adjustthe gap 302 radius a as desired (within limits, as described above). Forexample, an MCM 300 can be designed to obtain a specific OCRVsensitivity S_(VO), a specified dimension (e.g., gap 302 radius a, gap302 height t_(g) or radiation plate 310 thickness t_(m)), a specifiedbias voltage V_(DC), or a specified value for one or more other selectedvariables; while remaining within parametric ranges corresponding to anMCM 300 capable of maintaining linearly elastic, uncollapsed operation.

Advantageously, the design process can be initiated by choosing anormalized static mechanical force F_(b)/F_(g) and a relative biasvoltage V_(DC)/V_(C) which will make uncollapsed operation highlyrobust. Generally, the higher the normalized static mechanical forceF_(b)/F_(g) and relative bias voltage V_(DC)/V_(C), the higher thenormalized OCRV sensitivity of the MCM 300. For example, the OCRVsensitivity at

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) = \left( {0.9,0.9} \right)$

is almost 40 dB higher than the OCRV sensitivity at

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) = \left( {0.1,0.1} \right)$

(see Equations 2-4, 8, 12, 32, 35 and 38), holding other variablesconstant when the operating point parameters are changed as stated.Similarly, the minimum gap 302 radius a_(min) (as described above) willbe approximately 30 times larger at than at

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) = \left( {0.9,0.9} \right)$

than at

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) = \left( {0.1,0.1} \right)$

(see Equations 3 and 19-22), holding other variables constant when theoperating point parameters are changed as stated. However, generally,the lower the normalized static mechanical force F_(b)/F_(g) andrelative bias voltage V_(DC)/V_(C), the more stable the MCM 300 will beagainst static pressure variations, production tolerances, andvariations in bias voltage conditions. The design processes describedherein enable various types of design objectives to be met efficientlyand with effective MCM 300 performance results.

Note, however, that sensitivity will generally be poor for MCMs 300 withnormalized static mechanical force and relative bias voltage level

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) < {\left( {0.1,0.1} \right).}$

Also, MCMs 300 with normalized static mechanical force and relative biasvoltage level

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{DC}}{V_{C}}} \right) > \left( {0.85,0.9} \right)$

(respectively) will be prone to collapse.

FIG. 13 shows an example process 1300 for design of an MCM 300, startingwith a selected OCRV sensitivity, normalized static mechanical forceF_(b)/F_(g), and relative bias voltage V_(DC)/V_(C). Accordingly, instep 1302, a gap 302 pressure, an OCRV sensitivity, a normalized staticmechanical force F_(b)/F_(g), and a relative bias voltage levelV_(DC)/V_(C) are selected, and K is set to equal one (K=1, see Equations26-29). For example, for an MCM 300 with a gap 302 containing vacuum,these selections can comprise an OCRV sensitivity of −60 dB at SAP, anormalized static mechanical force F_(b)/F_(g)=0.7, and a relative biasvoltage level V_(DC)/V_(C)=0.7.

In step 1304, normalized dimensions, normalized OCRV sensitivity andbias voltage V_(DC) are determined. For K=1, Equations 19 and 20 can beused to determine that the normalized maximum ratio between the gap 302radius and the radiation plate 310 thickness

$\left( \frac{a}{t_{m}} \right)_{N_{—}\max} = {0.986.}$

As described above, gap 302 radius a and radiation plate 310 thicknesst_(m) are inversely related to the ratio between gap 302 radius andradiation plate thickness 310 a/t_(m). Therefore, an MCM 300 in which

$\left( \frac{a}{t_{m}} \right)_{N} = {\left( \frac{a}{t_{m}} \right)_{N_{—}\max}\left( {K = 1} \right)}$

is the smallest MCM 300 which satisfies the elastic linearity constraintand has the specified sensitivity when operating at the specifiednormalized static mechanical force F_(b)/F_(g)=0.7 and relative biasvoltage level V_(DC)/V_(C)=0.7 (in the described example), and thecorresponding bias voltage V_(DC) (the minimum bias voltage V_(DC) toproduce the specified OCRV sensitivity; as described above, increasingbias voltage V_(DC) increases OCRV sensitivity) That is, normalizeddimensions determined in step 1304 will be the minimum normalizeddimensions. Equations 8 and 22-24 can then be used to determine thesenormalized minimum dimensions: normalized gap 302 radius a_(n)=2.198;normalized radiation plate 310 thickness t_(m) _(_) _(n)=2.221;normalized effective gap 302 height t_(ge) _(_) _(n)=3.00; andnormalized OCRV sensitivity S_(VO)−20 log V_(DC) _(_) _(n)=63.64 dB. Thenormalized bias voltage V_(DC) _(_) _(n) and bias voltage V_(DC) can bedetermined using the normalized OCRV sensitivity S_(VO)−20 log V_(DC)_(_) _(n): V_(DC) _(_) _(n)=6.573×10⁻⁷ m and V_(DC)=47 V.

In step 1306, the dimensions determined in step 1304 are de-normalizedfor a selected radiation plate 310 material, to produce physicaldimensions of an MCM 300 with a vacuum gap 302 (the selected gappressure) and the selected sensitivity and operating point parameters.In the described example, the normalized dimensions correspond tode-normalized physical dimensions as follows (see Equations 8, 10, 13and 18): radiation plate 310 thickness t_(m)=10.42 μm, and effective gap302 height t_(ge)=2.82 μm; and for a crystalline silicon radiation plate310, with Young's modulus Y₀ of 250 GPa and Poisson's ratio σ of 0.14,gap 302 radius a=366.1 μm (in this example, input capacitance C_(in)=2.2pF). If the radiation plate 310 is made of a harder material, forexample a material with Young's modulus Y₀ of 250 GPa and Poisson'sratio σ of 0.14, gap 302 radius a=413 μm. Changing radiation plate 310hardness does not change radiation plate 310 thickness t_(m) oreffective gap 302 height t_(ge).

In step 1308, the gap 302 height t_(g) and the total insulator thicknesst_(i)=t_(i1)+t_(i2) of the first and second insulator layers 316, 316are determined from the effective gap 302 height t_(ge) usingEquation 1. The gap height t_(g) is preferably large enough to enable,with a margin, the radiation plate 310 to be displaced by the staticdisplacement of the center of the radiation plate 310 X_(P) without theradiation plate 310 collapsing. That is, a “safe” gap 302 height t_(g)should be chosen, meaning sufficient room should be given to compensatefor variations in operating conditions, such as variations in biasvoltage V_(DC) (and therefore relative bias voltage level V_(DC)/V_(C))due to variations in a voltage supply providing the bias voltage, orchanges in atmospheric pressure due to weather or pressure waves(sounds) incident on the radiation plate 310. In the described example,normalized static displacement

$\frac{X_{P}}{t_{ge}} = {{\frac{F_{b}}{F_{g}}\left( \frac{a}{t_{m}} \right)_{N_{—}\max}^{- 4}} = {{(0.7)(1.058)} = {0.74.}}}$

This results in static displacement X_(P)=2.09 μm. As determined above,effective gap height t_(ge)=2.82 μm. If t_(g)=2.50 μm is chosen as asafe gap 302 height, then the total insulator thickness ti is limited by

${\frac{t_{i\; 1}}{ɛ_{r\; \_ \; i\; 1}} + \frac{t_{i\; 2}}{ɛ_{r\; \_ \; i\; 2}}} = {{t_{ge} - t_{g}} = {{2.82 - 2.50} = {{.32}\mspace{14mu} {{\mu m}.}}}}$

If an insulator material is selected for both insulator layers 314, 316with a relative permittivity of 4, then total insulator thicknesst_(i)=1.28 μm.

In step 1310, to make linear elastic operation more robust, and if alarger gap 302 radius is desired (such as to improve SCRC sensitivity),then the scaling method described by equations 26-29 can be used.Accordingly, K can be selected (preferably, within describedlimitations) to adjust (increase) the gap 302 radius a and the radiationplate 310 thickness t_(m), and adjust (decrease) the ratio between thegap 302 radius and the radiation plate 310 thickness a/t_(m). Steps 1304through 1310 can then be repeated with the new K, until the desiredresults are achieved.

FIG. 14 shows an example process 1400 for design of an MCM 300, startingwith a selected normalized static mechanical force F_(b)/F_(g), biasvoltage V_(DC), and relative bias voltage V_(DC)/V_(C). In step 1402,the process begins by selecting a gap 302 pressure, a normalized staticmechanical force F_(b)/F_(g), a bias voltage V_(DC), and a relative biasvoltage level V_(DC)/V_(C); and by setting K to equal 1 (K=1, seeEquations 26-29). For example, for an MCM 300 with a gap 302 containingvacuum, these selections can comprise a bias voltage V_(DC)=10 volts, anormalized static mechanical force F_(b)/F_(g)=0.7, and a relative biasvoltage level V_(DC)/V_(C)=0 .7.

In step 1404, for K=1, Equations 19 and 20 can be used to determine thatthe normalized maximum ratio between the gap 302 radius and theradiation plate 310 thickness

$\left( \frac{a}{t_{m}} \right)_{N\; \_ \; {ma}\; x} = {0.986.}$

Also, Equation 12 can be used to determine V_(DC) _(_) _(n)=1.4×10⁻⁷ (m)at SAP. Normalized dimensions and other values thus correspond to:normalized gap 302 radius a_(n)=2.198, normalized radiation plate 310thickness t_(m) _(_) _(n)=2.221, normalized gap 302 height t_(ge) _(_)_(n)=3.00, and normalized OCRV sensitivity S_(VO)−20 log V_(DC) _(_)_(n)=63.64 dB (in this example, normalized input capacitance C_(in) _(_)_(n)=3.7890).

In step 1406, MCM 300 dimensions are de-normalized to determine physicaldimensions using a selected radiation plate 310 material. In thedescribed example, the normalized dimensions correspond to de-normalizedphysical dimensions as follows: radiation plate 310 thickness t_(m)=2.22μm (at K=1, this is a minimum radiation plate 310 thickness), effectivegap height t_(ge)=600 nm, and OCRV sensitivity

${S_{VO} = {{- 73.44}\mspace{14mu} {dB}\mspace{14mu} {re}\; \frac{V}{PA}}};$

and for a crystalline silicon radiation plate 310, gap 302 radius a=78.0μm and input capacitance C_(in)=0.5 pF (a relatively low capacitance foruse in conjunction with pre-amplification circuitry). Gap 302 heightt_(g) and total insulator thickness t_(i) can also be determined(similarly to step 1308).

In the example discussed with respect to FIG. 14, the resulting gap 302radius a=78.0 μm (aperture) is small, and will generally correspond to alow SCRC sensitivity. In some embodiments, amplification circuitry usedto amplify an MCM 300 output signal can result in a loss in total SNR ofthe microphone-plus-amplification system that reduces SNR below designspecifications. In some embodiments corresponding to the exampledescribed with respect to FIG. 14 (with gap 302 radius a=78.0 μm), amicrophone can comprise sufficient MCM 300 cells, electrically connectedin parallel (such as in a closed pack array), to cover an area of 1 mm²;for example, 30 such cells. The input capacitance of such an array wouldbe 30*0.5=15 pF.

Alternatively, in step 1408, to make linear elastic operation morerobust, and if a larger gap 302 radius is desired (such as to improveSCRC sensitivity), then the scaling method described by Equations 26-29can be used. Accordingly, K can be selected (preferably, withindescribed limitations) to adjust (increase) the gap 302 radius a and theradiation plate 310 thickness t_(m), and adjust (decrease) the ratiobetween the gap 302 radius and the radiation plate 310 thicknessa/t_(m). Steps 1404 through 1308 can then be repeated with the new K,until the desired results are achieved.

For example, K=2, meaning that

$\left( \frac{a}{t_{M}} \right)_{N} = {{\frac{1}{2}\left( \frac{a}{t_{m}} \right)_{N\; \_ \; m\; {ax}}} = {0.493.}}$

This results in normalized dimensions (see step 1404 and Equations26-29) as follows: normalized gap 302 radius a_(n)=(2³)a_(n) _(_)_(min)=17.584; and normalized radiation plate 310 thickness t_(m) _(_)_(n)=(2⁴)t_(m) _(_) _(n) _(_) _(min)=17.768. The physical dimensionsthen become, for a crystallized silicon radiation plate 310 (see step1406 and Equations 16 and 17): gap 302 radius a=626.0 μm; and radiationplate 310 thickness t_(m)=35.5 μm. Changes in K do not change effectivegap height t_(ge) or OCRV sensitivity S_(VO) (except with respect toeffects due to the elastic linearity constraint being an approximation).

FIG. 15 shows an example process 1500 for design of an MCM 300, startingwith a selected gap 302 radius a, bias voltage V_(DC), and radiationplate 310 material. Accordingly, in step 1502, a gap 302 pressure, a gap302 radius a, a bias voltage V_(DC), a radiation plate 310 material areselected. In step 1504, the specified bias voltage V_(DC) is used todetermine a range of normalized minimum gap 302 radii a_(n) _(_) _(min)corresponding to preferred

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{D\; C}}{V_{C}}} \right)$

pairs (such as a range of such pairs), along with corresponding OCRVsensitivities. In step 1506, the range of minimum gap 302 radii a_(n)_(_) _(min) is denormalized for the preferred material using thematerial's properties and the bias voltage V_(DC) to obtain a range ofphysical minimum gap 302 radii a_(min). In step 1506, if the specifiedgap 302 radius a is larger than a determined minimum gap 302 radiusa_(min) corresponding to a preferred

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{D\; C}}{V_{C}}} \right)$

pair and corresponding determined OCRV sensitivity, then the minimum gap302 radius a_(min) can be scaled up to the specified gap 302 radius ausing (and as described with respect to) Equations 26-29. In step 1508,other MCM 300 dimensions can be determined using the selected operatingpoint, the normalized dimensions at the selected operating point, andthe scaling constant K determined in step 1506.

FIG. 16 shows an example process 1600 for design of an MCM 300, startingwith a selected radiation plate 310 thickness t_(m) or effective gap 302height t_(ge), and a selected bias voltage V_(DC) or other parameterdependent on bias voltage V_(DC). (For example, OCRV sensitivity isdependent on bias voltage V_(DC).) In step 1602, a gap 302 pressure, aradiation plate 310 thickness t_(m) or effective gap 302 height t_(ge),a bias voltage V_(DC), a radiation plate 310 material are selected. Instep 1604, the specified dimension is normalized for the selected biasvoltage V_(DC), or for another parameter (such as sensitivity) which isdependent on bias voltage V_(DC). In step 1604, the normalized staticmechanical force and relative bias voltage level

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{D\; C}}{V_{C}}} \right)$

corresponding to this normalized dimension are then determined. In step1606, a safe operating point is chosen (similarly to step 1308) from thedetermined parameters, and other physical dimensions suitable forproduction using a preferred material are determined (including anydesired rescaling).

The design processes described in FIGS. 13-16 are merely exemplary;other design processes can be used, as described herein.

FIG. 17 shows a graph 1700 of the relationship between normalizedminimum gap 302 radius a_(n) _(_) _(min) and normalized effective gap302 height t_(ge) _(_) _(n) for various values of the relative biasvoltage level V_(DC)/V_(C). As described above, sensitivity willgenerally be poor for MCMs 300 with normalized static mechanical forceand relative bias voltage level

${\left( {\frac{F_{b}}{F_{g}},\frac{V_{D\; C}}{V_{C}}} \right) < \left( {0.1,0.1} \right)};$

and MCMs 300 with normalized static mechanical force and relative biasvoltage level

$\left( {\frac{F_{b}}{F_{g}},\frac{V_{D\; C}}{V_{C}}} \right) > \left( {0.85,0.9} \right)$

(respectively) will be prone to collapse. FIG. 17 shows curves relatingthe normalized minimum gap 302 radius a_(n) _(_) _(min) to thenormalized effective gap 302 height t_(ge) _(_) _(n). A first curve 1702corresponds to the described upper (0.9) bound for the relative biasvoltage level

$\frac{V_{D\; C}}{V_{C}}.$

A second curve 1704 corresponds to the described lower (0.1) bound forthe relative bias voltage level

$\frac{V_{D\; C}}{V_{C}}.$

Individual curves correspond to varying normalized static mechanicalforce

$\frac{F_{b}}{F_{g}}.$

lne dotted-line boundary 1706 corresponds (with a small margin) to thex-axis and y-axis limits reached by the respective curves 1702, 1704 atthe described upper (0.85) and lower (0.1) bounds for the normalizedstatic mechanical force

$\frac{F_{b}}{F_{g}}.$

The dotted-line boundary 1706 corresponds to an area within the graph1700 which is enclosed by the curves and ranges (of the normalizedeffective gap 302 height t_(ge) _(_) _(n)) described by Equations 50 and51.

a _(n) _(_) _(min)≤0.9t _(ge) _(_) _(n) for t _(ge) _(_) _(n) in therange 0.2<t _(ge) _(_) _(n)≤0.8   Equation 50

0.9t _(ge) _(_) _(n)−0.72<a _(n) _(_) _(min)≤0.9t _(ge) _(_) _(n) for t_(ge) _(_) _(n) in the range 0.8<t _(ge) _(_) _(n)≤6.8   Equation 51

The dotted-line boundary 1706 closely approximates the first curve 1702and the second curve 1704. As previously stated, the elastic linearityconstraint is an approximation; portions of the described regionsbeneath the curves (lower normalized gap 302 radius a_(n) than thenormalized minimum gap 302 radius a_(n) _(_) _(min)) remain close enoughto the boundaries set by the elastic linearity constraint to enable, insome embodiments, useful operation with small reductions in OCRVsensitivity with respect to the elastic linear operation region (asdescribed above with respect to the elastic linearity constraintboundary).

FIG. 18 shows a graph 1800 of the relationship between normalizedminimum radiation plate 310 thickness t_(m) _(_) _(n) _(_) _(min) andnormalized effective gap 302 height t_(ge) _(_) _(n) for various valuesof the relative bias voltage level V_(DC)/V_(C). FIG. 18 shows curvesrelating the normalized minimum radiation plate 310 thickness t_(m) _(_)_(n) _(_) _(min) to the normalized effective gap 302 height t_(ge) _(_)_(n). A first curve 1802 corresponds to the described upper (0.9) boundfor the relative bias voltage level

$\frac{V_{D\; C}}{V_{C}}.$

A second curve 1804 corresponds to the described lower (0.1) bound forthe relative bias voltage level

$\frac{V_{D\; C}}{V_{C}}.$

Individual curves correspond to varying normalized static mechanicalforce

$\frac{F_{b}}{F_{g}}.$

The dotted-line boundary 1806 corresponds (with a small margin) to thex-axis and y-axis limits reached by the respective curves 1802, 1804 atthe described upper (0.85) and lower (0.1) bounds for the normalizedstatic mechanical force

$\frac{F_{b}}{F_{g}}.$

The dotted-line boundary 1806 corresponds to an area within the graph1800 which is enclosed by the curves and ranges (of the normalizedeffective gap 302 height t_(ge) _(_) _(n)) described by Equations 52 and53.

t _(m) _(_) _(n) _(_) _(min)≤0.93t _(ge) _(_) _(n)−0.186 for t _(ge)_(_) _(n) in the range 0.2<t _(ge) _(_) _(n)≤0.8   Equation 52

0.93t _(ge) _(_) _(n)−0.744<t _(m) _(_) _(n) _(_) _(min)≤0.93t _(ge)_(_) _(n)−0.186 for t _(ge) _(_) _(n) in the range 0.8<t _(ge) _(_)_(n)≤6.8    Equation 53

The dotted-line boundary 1806 closely approximates the first curve 1802and the second curve 1804. As previously stated, the elastic linearityconstraint is an approximation; portions of the described regionsbeneath the curves (lower normalized radiation plate 310 thickness t_(m)_(_) _(n) than the normalized minimum radiation plate 310 thicknesst_(m) _(_) _(n) _(_) _(min)) remain close enough to the boundaries setby the elastic linearity constraint to enable, in some embodiments,useful operation with small reductions in OCRV sensitivity with respectto the elastic linear operation region (as described above with respectto the elastic linearity constraint boundary).

FIG. 19 shows a graph 1900 of the relationship between normalized gap302 radius a_(n) and normalized effective gap 302 height t_(ge) _(_)_(n) for various values of the relative bias voltage level V_(DC)/V_(C)and various values of the scaling constant K. Specifically, FIG. 19applies the scaling constant K=2.5 to the first curve 1702

$\left( {\frac{V_{DC}}{V_{C}} = {.9}} \right)$

of FIG. 17 to produce a first scaled curve 1902, and applies the scalingconstant K=2.5 to the second curve 1704

$\left( {\frac{V_{DC}}{V_{C}} = {.1}} \right)$

of FIG. 17 to produce a second scaled curve 1904 (see Equations 26-29).The dotted-line boundary 1906 corresponds (with a small margin) to thex-axis and y-axis limits reached by the respective curves 1902, 1904 atthe described upper (0.85) and lower (0.1) bounds for the normalizedstatic mechanical force

$\frac{F_{b}}{F_{g}}$

in FIG. 17. (The dotted-line boundary 2006 overlaps the second scaledcurve 1904.) The dotted-line boundary 1906 closely approximates anddescribes a useful region of MCM 300 normalized dimensions foruncollapsed, linear elastic operation. The dotted-line boundarycorresponds to an area within the graph 1900 which is enclosed by thecurves and ranges (of the normalized effective gap 302 height t_(ge)_(_) _(n)) described by Equations 54 and 55.

a _(n)≤14.2t _(ge) _(_) _(n)−2.84 for t _(ge) _(_) _(n) in the range0.2<t _(ge) _(_) _(n)≤0.8   Equation 54

0.9t _(ge) _(_) _(n)−0.72<a _(n)≤14.2t _(ge) _(_) _(n)−2.84 for t _(ge)_(_) _(n) in the range 0.8<t _(ge) _(_) _(n)≤6.8   Equation 55

FIG. 20 shows a graph 2000 of the relationship between normalizedradiation plate 310 thickness t_(m) _(_) _(n) and normalized effectivegap 302 height t_(ge) _(_) _(n) for various values of the relative biasvoltage level V_(DC)/V_(C) and various values of the scaling constant K.Specifically, FIG. 20 applies the scaling constant K=2.5 to the firstcurve 1802

$\left( {\frac{V_{DC}}{V_{C}} = {.9}} \right)$

of FIG. 18 to produce a first scaled curve 2002, and applies the scalingconstant K=2.5 to the second curve 1804

$\left( {\frac{V_{DC}}{V_{C}} = {.1}} \right)$

of FIG. 18 to produce a second scaled curve 2004 (see Equations 26-29).The dotted-line boundary 2006 corresponds (with a small margin) to thex-axis and y-axis limits reached by the respective curves 2002, 2004 atthe described upper (0.85) and lower (0.1) bounds for the normalizedstatic mechanical force

$\frac{F_{b}}{F_{g}}$

in FIG. 18. (The dotted-line boundary 2006 overlaps the second scaledcurve 2004.) The dotted-line boundary 2006 closely approximates anddescribes a useful region of MCM 300 normalized dimensions foruncollapsed, linear elastic operation. The dotted-line boundarycorresponds to an area within the graph 2000 which is enclosed by thecurves and ranges (of the normalized effective gap 302 height t_(ge)_(_) _(n)) described by Equations 56 and 57.

t _(m) _(_) _(n)≤36t _(ge) _(_) _(n)−7.2 for t _(ge) _(_) _(n) in therange 0.2<t _(ge) _(_) _(n)≤0.8   Equation 56

0.93t _(ge) _(_) _(n)−0.744<t _(m) _(_) _(n)≤36t _(ge) _(_) _(n)−7.2 fort _(ge) _(_) _(n) in the range 0.8<t _(ge) _(_) _(n)≤6.8   Equation 57

The disclosed innovations, in various embodiments, provide one or moreof at least the following advantages. However, not all of theseadvantages result from every one of the innovations disclosed, and thislist of advantages does not limit the variously claimed inventive scope.

-   -   Microphone dimensions for optimal microphone sensitivity can be        specified using a limited number of selected operating        parameters and/or dimensions and/or other microphone        characteristics;    -   uses a sealed gap, avoiding gap contamination;    -   sealed gap enables microphone operation, without damage to the        microphone, down to tens of meters under water;    -   self-noise of an MCM is limited to radiation impedance, so that        SNR is approximately 94 dBA;    -   suitable for use in various airborne consumer and professional        products, such as computers, ear phones, hearing aids, mobile        phones, wireless equipment and wideband precision acoustic        measurement and recording systems;    -   can be fabricated at low cost using standard MEMS processes;    -   microphone dimensions, sensitivity and other performance        characteristics are independent of materials used to fabricate        the radiation plate and insulator layers; and    -   avoids use of finite element analysis to optimize microphone        dimensions.

Sealed gap capacitive MEMS microphone embodiments, as disclosed herein,has very low self-noise, and can be designed for robust uncollapsed,linear elastic operation with high (or optimal) OCRV sensitivity. Theinventors have discovered that MCM performance (sensitivity) depends ona small number of operating parameters: static mechanical force, biasvoltage, and relative bias voltage level. These parameters—or dimensionsor other microphone properties dependent on these parameters—can bespecified at the start of a design process. This enables a sort ofdesign-in-reverse, allowing a designer to pick a desired performanceprofile of an MCM; microphone dimensions (gap radius/radiation plateradius, radiation plate thickness, and gap height) and othercharacteristics of the MCM are then determined by the selectedperformance profile. Radiation plate dimensions can then be scaled toimprove robustness of linearly elastic, uncollapsed operation, and toimprove SCRC sensitivity. Generally, these microphones are as durablewith respect to temperature and impact as pressure compensated MEMSmicrophones. Further, these microphones can be manufactured using toolsand processes used to manufacture pressure compensated MEMS microphones,making manufacture relatively inexpensive.

Modifications and Variations

As will be recognized by those skilled in the art, the innovativeconcepts described in the present application can be modified and variedover a tremendous range of applications, and accordingly the scope ofpatented subject matter is not limited by any of the specific exemplaryteachings given. It is intended to embrace all such alternatives,modifications and variations that fall within the spirit and broad scopeof the appended claims.

While certain variables are described herein as depending “only” oncertain other variables, this convention explicitly ignores variationsin as-fabricated parts, such as variations due to process variability,variations in process environment or operational environment, and otherfactors not addressed herein. These factors will generally not affectthe optimality of results with respect to particular operating points,as described herein.

In some embodiments, an MCM comprises an electret. In some MCMembodiments using an electret, the radiation plate can comprise apolymeric material.

The Electret and Performance reference shows that an electret layer inan MCM results in a DC bias voltage V_(E) that adds to the electricallyinduced bias voltage V_(DC), resulting in a total bias voltage ofV_(DC)+V_(E). The magnitude and polarity of effective electret voltageV_(E) depend on the polarization of the trapped charges in the electretlayer(s). When there is no external bias voltage, i.e. V_(DC)=0 volts, astatic bias is provided by V_(E) if the electrical termination isappropriate. This is particularly useful in transducer receptionapplications. Increased effective bias voltage as a result of anelectret can be used to increase the sensitivity of the MCM.

In some embodiments, a membrane is used as a vibrating element.

In some embodiments, ambient pressure can be taken to be between 70 kPa,corresponding to approximately the lowest normal pressure in an airplanecabin, and 110 kPa, corresponding to a highest atmospheric pressuremeasured on Earth.

In some embodiments, an MCM uses a single insulator layer of thicknesst_(i)=t_(i1)+t_(i2).

In some embodiments, an MCM with amplification can achieve asignal-to-noise ratio of 75 dB or more.

In some embodiments, an electret is used in addition to or instead of anapplied bias voltage.

In some embodiments, an MCM scaled pursuant to Equations 26-29 will haveSCRC sensitivity K⁶ times greater than an unscaled MCM.

In some embodiments using a number N MCMs electrically connected inparallel, the connected MCMs together have N times greater SCRCsensitivity than a single one of the MCMs.

While “optimum” sensitivity and maintaining “optimum” sensitivity (orother determined sensitivity) are referred to herein, one of ordinaryskill in the arts of capacitive MEMS microphones will understand thatfabrication tolerances, variations in the static pressure differencebetween the ambient and the gap (such as between the Dead Sea andLhasa), material imperfections causing variations of material elasticproperties, variations from the operating point during operation, theapproximate nature of the elastic linearity constraint, and otherdifferences between models and physicalized embodiments can causevariation of an MCM's sensitivity from the “optimum” sensitivity.

In some embodiments, the operating point is selected by selecting up tothree of the following: the gap radius a, the radiation plate thicknesst_(m), the effective gap height t_(ge), the optimum OCRV sensitivity, anSCRC sensitivity, the normalized static mechanical force F_(b)/F_(g),the bias voltage V_(DC), and the relative bias voltage levelV_(DC)/V_(C).

In some embodiments, MCM microphones can be connected in parallel toyield the same OCRV sensitivity as a single element, but with higherSCRC sensitivity and higher input capacitance.

In some embodiments, MCM microphones can be connected in parallel toyield higher OCRV sensitivity and lower SCRC sensitivity and inputcapacitance.

Additional general background, which helps to show variations andimplementations, may be found in the following publications, all ofwhich are hereby incorporated by reference: U.S. Pat. No. 6,075,867;U.S. Pat. No. 7,955,250; U.S. Pat. No. 8,288,971; U.S. Pat. No.9,363,589; U.S. Pat. No. 9,451,375; U.S. Pat. No. 9,560,430; U.S. Pat.Pub. No. 2001/0019945; U.S. Pat. Pub. 2014/0339657; U.S. Pat. Pub. No.2014/0083296; and U.S. Pat. Pub. No. 2015/0163572; H. Köymen, A. Atalar,E. Aydoğdu, C. Kocabas̨, H. K. Oğuz, S. Olçum, A. Özgürlük, A. Ünlügedik,“An improved lumped element nonlinear circuit model for a circular CMUTcell,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 59, no. 8,pp. 1791-1799, August 2012; H. Köymen, A. Atalar, I. Köymen, A. S.Tas̨delen, A. Ünlügedik, “Unbiased Charged Circular CMUT Microphone:Lumped Element Modeling and Performance”, IEEE Trans. Ultrason.Ferroelectr. Freq. Control, Vol. 65, no. 1, pp. 60-71, Nov. 14, 2017; A.Unlugedik, A. S. Tasdelen, A. Atalar, and H. Koymen, “DesigningTransmitting CMUT Cells for Airborne Applications,” IEEE Trans.Ultrason. Ferroelectr. Freq. Control, Vol. 61, pp. 1899-1910, 2014; M.Funding la Cour, T. L. Christiansen, J. A. Jensen, and E. V. Thomsen,“Electrostatic and Small-Signal Analysis of CMUTs With Circular andSquare Anisotropic Plates,” IEEE Trans. Ultrason. Ferroelectr. Freq.Control, vol. 62, no. 8, pp. 1563-1579, 2015; H. Köymen, A. Atalar andH. K. Oğuz, “Designing Circular CMUT Cells Using CMUT Biasing Chart,”2012 IEEE International Ultrasonics Symposium Proceedings pp. 975-978,Dresden, October, 2012; M. Engholm, T. Pedersen, and E.V. Thomsen,“Modeling of plates with multiple anisotropic layers and residualstress,” Sens. and Act. A: Phys., vol. 240, pp. 70-79, April 2016; andM. Rahman, J. Hernandez, S. Chowdhury, “An Improved Analytical Method toDesign CMUTs With Square Diaphraghms,” IEEE Trans. Ultrason.Ferroelectr. Freq. Control, vol. 260, no. 4, April 2013.

None of the description in the present application should be read asimplying that any particular element, step, or function is an essentialelement which must be included in the claim scope: THE SCOPE OF PATENTEDSUBJECT MATTER IS DEFINED ONLY BY THE ALLOWED CLAIMS. Moreover, none ofthese claims are intended to invoke paragraph six of 35 USC section 112unless the exact words “means for” are followed by a participle.

The claims as filed are intended to be as comprehensive as possible, andNO subject matter is intentionally relinquished, dedicated, orabandoned.

As shown and described herein, the inventors have discovered a varietyof new and useful approaches to capacitive MEMS microphones with asealed gap, and design of such microphones.

1. A microphone system for receiving sound waves, the microphone systemcomprising: a back plate; a radiation plate having a thickness t_(m),the radiation plate clamped to the back plate so that there is a sealedgap between the radiation plate and the back plate such that passage ofgas into or out of the gap is prevented, the gap having a radius a and agap height t_(g); a first electrode, either the first electrode beingfixedly coupled to a side of the back plate proximate to the gap, or thefirst electrode comprising or contained within the back plate; a secondelectrode, either the second electrode being fixedly coupled to a sideof the radiation plate, or the first electrode comprising or containedwithin the radiation plate; a first insulator layer of thickness t_(i1)and relative permittivity ε_(r) _(_) _(i1), and a second insulator layerof thickness t_(i2) and relative permittivity ε_(r) _(_) _(i2), thefirst and second insulator layers being disposed between the first andsecond electrodes, and the first and second insulator layers beingdisposed between the back plate and the radiation plate; a power source;and a microphone controller configured to use the power source to drivethe microphone at an operating point, wherein F_(b) is a net staticforce exerted on the radiation plate due to an ambient static pressure,F_(g) is a uniformly distributed force required to displace a center ofthe radiation plate by an effective gap height t_(ge), and V_(C) is alimit to bias voltage V_(DC) for uncollapsed operation of the microphonesystem, the operating point comprising: a normalized static mechanicalforce F_(b)/F_(g), a bias voltage of the first and second electrodesV_(DC), and a relative bias voltage level of the first and secondelectrodes V_(DC)/V_(C); wherein${t_{ge} = {t_{g} + \frac{t_{i\; 1}}{ɛ_{{r\_ i}1}} + \frac{t_{i\; 2}}{ɛ_{{r\_ i}2}}}};$and wherein the gap radius a, the gap height t_(g), and the radiationplate thickness t_(m) are determined using the selected operating pointso that an OCRV sensitivity of the microphone at the selected operatingpoint is an optimum OCRV sensitivity for the selected operating point.2. The microphone system of claim 1, wherein the gap comprises a holemachined into the substrate, and the back plate comprises a portion ofthe substrate forming a floor of the gap.
 3. The microphone system ofclaim 1, wherein the first electrode covers at least 80% of the area ofthe back plate on the side of the back plate proximate to the gap, andwherein the second electrode covers at least 80% of the area of theradiation plate on the side of the radiation plate proximate to the gap.4. The microphone system of claim 1, wherein the sound waves arehuman-audible and the gap contains a vacuum.
 5. The microphone system ofclaim 1, wherein the gap radius a, the gap height t_(g), and theradiation plate thickness t_(m) are determined using the operating pointso that the microphone system will maintain uncollapsed, linear elasticoperation.
 6. The microphone system of claim 1, wherein the gap iscircular.
 7. The microphone system of claim 1, further comprising anelectret configured to increase an effective bias voltage of the firstand second electrodes.
 8. The microphone system of claim 1, wherein theradiation plate comprises a selected solid material suitable forfabrication of a MEMS microphone; and wherein the particular selectedsolid material does not affect the optimum sensitivity, and does notaffect a corresponding gap height or radiation plate thickness.
 9. Themicrophone system of claim 1, wherein the gap radius a is related to aminimum gap radius a_(min) corresponding to the optimum sensitivity atthe operating point, and the radiation plate thickness t_(m) is relatedto a minimum radiation plate thickness t_(m) _(_) _(min) correspondingto the optimum sensitivity at the operating point, by a selected scalingconstant K, such that a=(K³)a_(min), and t_(m)=(K⁴)t_(m) _(_) _(min).10. The microphone system of claim 1, wherein the operating point is aselected operating point, the selected operating point being selected byselecting up to three of the following: the gap radius a, the radiationplate thickness t_(m), the effective gap height t_(ge), the optimum OCRVsensitivity, an SCRC sensitivity, the normalized static mechanical forceF_(b)/F_(g), the bias voltage V_(DC), and the relative bias voltagelevel V_(DC)/V_(C).
 11. The microphone system of claim 1, whereinmultiple ones of the microphone system are electrically connected inparallel.
 12. A microphone system for receiving sound waves, themicrophone system comprising: a back plate; a radiation plate having athickness t_(m), the radiation plate clamped to the back plate so thatthere is a sealed gap between the radiation plate and the back platesuch that passage of gas into or out of the gap is prevented, the gaphaving a radius a and a gap height t_(g); a first electrode, either thefirst electrode being fixedly coupled to a side of the back plateproximate to the gap, or the first electrode comprising or containedwithin the back plate; a second electrode, either the second electrodebeing fixedly coupled to a side of the radiation plate, or the firstelectrode comprising or contained within the radiation plate; a firstinsulator layer of thickness t_(i1) and relative permittivity, ε_(r)_(_) _(i1), and a second insulator layer of thickness t_(i2) andrelative permittivity ε_(r) _(_) _(i2), the first and second insulatorlayers being disposed between the first and second electrodes, and thefirst and second insulator layers being disposed between the back plateand the radiation plate; a power source; and a microphone controllerconfigured to use the power source to drive the microphone at anoperating point, wherein F_(b) is a net static force exerted on theradiation plate due to an ambient static pressure, F_(g) is a uniformlydistributed force required to displace a center of the radiation plateby an effective gap height t_(ge), and V_(C) is a limit to bias voltageV_(DC) for uncollapsed operation of the microphone system, the operatingpoint comprising: a normalized static mechanical force F_(b)/F_(g), abias voltage of the first and second electrodes V_(DC), and a relativebias voltage level of the first and second electrodes V_(DC)/V_(C);wherein${t_{ge} = {t_{g} + \frac{t_{i\; 1}}{ɛ_{{r\_ i}1}} + \frac{t_{i\; 2}}{ɛ_{{r\_ i}2}}}};$wherein a_(n) is a normalized radius of the gap, and a_(n) is in therange:a _(n)≤14.2t _(ge) _(_) _(n)−2.84 for 0.2<t _(ge) _(_) _(n)≤0.80.9t _(ge) _(_) _(n)−0.72<a _(n)≤14.2t _(ge) _(_) _(n)−2.84 for 0.8<t_(ge) _(_) _(n)≤6.8; wherein t_(m) _(_) _(n) is a normalized thicknessof the radiation plate, and t_(m) _(_) _(n) is in the range:t _(m) _(_) _(n)≤36t _(ge) _(_) _(n)−7.2 for 0.2<t _(ge) _(_) _(n)≤0.80.93t _(ge) _(_) _(n)−0.744<t _(m) _(_) _(n)≤36t _(ge) _(_) _(n)−7.2 for0.8<t _(ge) _(_) _(n)≤6.8; wherein ε₀ is a permittivity of free space,P₀ is a static pressure difference between an ambient and the gap, andV_(DC) _(_) _(n) is a normalized operating bias voltage such that:${V_{DC\_ n} = {\frac{3}{2}\sqrt{\frac{ɛ_{0}}{P_{0}}}V_{DC}}};$wherein t_(ge) _(_) _(n) is a normalized effective gap height, and thegap radius a, the gap height t_(g), and the radiation plate thicknesst_(m) are:$\mspace{20mu} {t_{ge} = {{V_{DC\_ n}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}t_{ge\_ n}}}$${t_{ge\_ n}\left( \frac{F_{b}}{F_{g}} \right)} \approx \frac{\sqrt{\frac{F_{b}}{F_{g}}}}{0.9961 - {1.0468\frac{F_{b}}{F_{g}}} + {0.06972\left( {\frac{F_{b}}{F_{g}} - 0.25} \right)^{2}} + {0.01148\left( \frac{F_{b}}{F_{g}} \right)^{6}}}$$\mspace{20mu} {a = {\left( {10^{4}\sqrt{\frac{Y_{0}}{15\left( {1 - \sigma^{2}} \right)P_{0}}}} \right)V_{DC\_ n}a_{n}}}$$\mspace{20mu} {{t_{m} = {5{V_{DC\_ n}\left( \frac{V_{DC}}{V_{C}} \right)}^{- 1}t_{m\_ n}}};}$and wherein Y₀ is a Young's modulus of a material comprising theradiation plate and σ is a Poisson's ratio of the material comprisingthe radiation plate.
 13. The microphone system of claim 12, wherein thenormalized gap radius a_(n) corresponds to a normalized minimum gapradius a_(n) _(_) _(min) that is within the range for a_(n), thenormalized radiation plate thickness t_(m) _(_) _(n) corresponds to anormalized minimum radiation plate thickness t_(m) _(_) _(n) _(_) _(min)that is within the range for t_(m) _(_) _(n), K is a selected scalingconstant, X_(P) is a static deflection of the center of the radiationplate, $g\left( \frac{X_{P}}{t_{ge}} \right)$ is a function of$\frac{X_{P}}{t_{ge}},{{and}\mspace{14mu} {g^{\prime}\left( \frac{X_{P}}{t_{ge}} \right)}}$is a function which is the first derivative of$\mspace{20mu} {{g\left( \frac{X_{P}}{t_{ge}} \right)}\text{:}}$$\mspace{20mu} {{g\left( \frac{X_{P}}{t_{ge}} \right)} = \frac{\tanh^{- 1}\left( \sqrt{X_{P}/t_{ge}} \right)}{\sqrt{X_{P}/t_{ge}}\;}}$$\mspace{20mu} {{g^{\prime}\left( \frac{X_{P}}{t_{ge}} \right)} = {\frac{1}{2\frac{X_{P}}{t_{ge}}}\left( {\frac{1}{1 - \frac{X_{P}}{t_{ge}}} - {g\left( \frac{X_{P}}{t_{ge}} \right)}} \right)}}$$\mspace{20mu} {\frac{a_{n\_ min}}{t_{{m\_ n}{\_ min}}} = \sqrt[4]{\frac{F_{b}}{F_{g}}/\frac{X_{P}}{t_{ge}}}}$${{\frac{V_{DC}^{2}}{V_{C}^{2}}\left( {0.9961 - {1.0468\frac{F_{b}}{F_{g}}} + {0.06972\left( {\frac{F_{b}}{F_{g}} - 0.25} \right)^{2}} + {0.01148\left( \frac{F_{b}}{F_{g}} \right)^{6}}} \right)^{2}2{g^{\prime}\left( \frac{X_{P}}{t_{ge}} \right)}} - {3\left( {\frac{X_{P}}{t_{ge}} - \frac{F_{b}}{F_{g}}} \right)}} \approx 0$$\mspace{20mu} {{{for}\mspace{14mu} \frac{X_{P}}{t_{ge}}} \geq \frac{F_{b}}{F_{g}}}$  a_(n) = (K³)a_(n_min)   t_(m_n) = (K⁴)t_(m_n_min).
 14. Themicrophone system of claim 12, wherein the material comprising theradiation plate is a selected solid material suitable for fabrication ofa MEMS microphone; and wherein the particular selected solid materialdoes not affect the optimum sensitivity, and does not affect acorresponding gap height or radiation plate thickness.
 15. Themicrophone system of claim 12, wherein both insulator layers are fixedlycoupled to the radiation plate, or both insulator layers are fixedlycoupled to the back plate, or the first insulator layer is fixedlycoupled to the radiation plate and the second insulator layer is fixedlycoupled to the back plate.
 16. The microphone system of claim 12,wherein the gap comprises a hole machined into the substrate, and theback plate comprises a portion of the substrate forming a floor of thegap.
 17. The microphone system of claim 12, wherein the first electrodecovers at least 80% of the area of the back plate on the side of theback plate proximate to the gap, and wherein the second electrode coversat least 80% of the area of the radiation plate on the side of theradiation plate proximate to the gap.
 18. The microphone system of claim12, wherein the sound waves include the range of human-audible soundwaves, and the gap contains a vacuum.
 19. The microphone system of claim12, wherein the gap radius a, the gap height t_(g), and the radiationplate thickness t_(m) are determined using the operating point so thatthe microphone system will maintain uncollapsed, linear elasticoperation.
 20. The microphone system of claim 12, wherein the gap iscircular.
 21. The microphone system of claim 12, further comprising anelectret configured to increase an effective bias voltage of the firstand second electrodes.
 22. (canceled)
 23. The microphone system of claim12, wherein the gap radius a is related to a minimum gap radius a_(min)corresponding to an optimum sensitivity at the operating point, and theradiation plate thickness t_(m) is related to a minimum radiation platethickness t_(m) _(_) _(min), corresponding to an optimum sensitivity atthe operating point, by a selected scaling constant K, such thata=(K³)a_(min), and t_(m)=(K⁴)t_(m) _(_) _(min).
 24. The microphonesystem of claim 12, wherein the operating point is a selected operatingpoint, the selected operating point being selected by selecting up tothree of the following: the gap radius a, the radiation plate thicknesst_(m), the effective gap height t_(ge), the optimum OCRV sensitivity, anSCRC sensitivity, the normalized static mechanical force F_(b)/F_(g),the bias voltage V_(DC), and the relative bias voltage levelV_(DC)/V_(C).
 25. The microphone system of claim 12, wherein multipleones of the microphone system are electrically connected in parallel.26-41. (canceled)